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Theorem cfsmolem 9406
Description: Lemma for cfsmo 9407. (Contributed by Mario Carneiro, 28-Feb-2013.)
Hypotheses
Ref Expression
cfsmolem.2 𝐹 = (𝑧 ∈ V ↦ ((𝑔‘dom 𝑧) ∪ 𝑡 ∈ dom 𝑧 suc (𝑧𝑡)))
cfsmolem.3 𝐺 = (recs(𝐹) ↾ (cf‘𝐴))
Assertion
Ref Expression
cfsmolem (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
Distinct variable groups:   𝑓,𝑔,𝑡,𝑤,𝑧,𝐴   𝑓,𝐹,𝑡,𝑧   𝑓,𝐺,𝑤,𝑧
Allowed substitution hints:   𝐹(𝑤,𝑔)   𝐺(𝑡,𝑔)

Proof of Theorem cfsmolem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cff1 9394 . 2 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)))
2 cfon 9391 . . . . . . . . . . . 12 (cf‘𝐴) ∈ On
32oneli 6069 . . . . . . . . . . 11 (𝑥 ∈ (cf‘𝐴) → 𝑥 ∈ On)
433ad2ant3 1171 . . . . . . . . . 10 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
5 eleq1w 2888 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥 ∈ (cf‘𝐴) ↔ 𝑦 ∈ (cf‘𝐴)))
653anbi3d 1572 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ↔ (𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴))))
7 fveq2 6432 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
87eleq1d 2890 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐺𝑥) ∈ 𝐴 ↔ (𝐺𝑦) ∈ 𝐴))
96, 8imbi12d 336 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) ∈ 𝐴) ↔ ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ 𝐴)))
10 simpl1 1248 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → 𝑔:(cf‘𝐴)–1-1𝐴)
11 simpl2 1250 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → 𝐴 ∈ On)
12 ontr1 6008 . . . . . . . . . . . . . . . . . 18 ((cf‘𝐴) ∈ On → ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴)))
132, 12ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴))
1413ancoms 452 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → 𝑦 ∈ (cf‘𝐴))
15143ad2antl3 1244 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → 𝑦 ∈ (cf‘𝐴))
16 pm2.27 42 . . . . . . . . . . . . . . 15 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ 𝐴) → (𝐺𝑦) ∈ 𝐴))
1710, 11, 15, 16syl3anc 1496 . . . . . . . . . . . . . 14 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ 𝐴) → (𝐺𝑦) ∈ 𝐴))
1817ralimdva 3170 . . . . . . . . . . . . 13 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ 𝐴) → ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴))
19 cfsmolem.3 . . . . . . . . . . . . . . . . . . . 20 𝐺 = (recs(𝐹) ↾ (cf‘𝐴))
2019fveq1i 6433 . . . . . . . . . . . . . . . . . . 19 (𝐺𝑥) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥)
21 fvres 6451 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (cf‘𝐴) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥) = (recs(𝐹)‘𝑥))
2220, 21syl5eq 2872 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) = (recs(𝐹)‘𝑥))
23 recsval 7765 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → (recs(𝐹)‘𝑥) = (𝐹‘(recs(𝐹) ↾ 𝑥)))
24 recsfnon 7764 . . . . . . . . . . . . . . . . . . . . . . . 24 recs(𝐹) Fn On
25 fnfun 6220 . . . . . . . . . . . . . . . . . . . . . . . 24 (recs(𝐹) Fn On → Fun recs(𝐹))
2624, 25ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 Fun recs(𝐹)
27 vex 3416 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥 ∈ V
28 resfunexg 6734 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun recs(𝐹) ∧ 𝑥 ∈ V) → (recs(𝐹) ↾ 𝑥) ∈ V)
2926, 27, 28mp2an 685 . . . . . . . . . . . . . . . . . . . . . 22 (recs(𝐹) ↾ 𝑥) ∈ V
30 dmeq 5555 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (recs(𝐹) ↾ 𝑥) → dom 𝑧 = dom (recs(𝐹) ↾ 𝑥))
3130fveq2d 6436 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑔‘dom 𝑧) = (𝑔‘dom (recs(𝐹) ↾ 𝑥)))
32 fveq1 6431 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑧𝑡) = ((recs(𝐹) ↾ 𝑥)‘𝑡))
33 suceq 6027 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑧𝑡) = ((recs(𝐹) ↾ 𝑥)‘𝑡) → suc (𝑧𝑡) = suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
3432, 33syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (recs(𝐹) ↾ 𝑥) → suc (𝑧𝑡) = suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
3530, 34iuneq12d 4765 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (recs(𝐹) ↾ 𝑥) → 𝑡 ∈ dom 𝑧 suc (𝑧𝑡) = 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
3631, 35uneq12d 3994 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (recs(𝐹) ↾ 𝑥) → ((𝑔‘dom 𝑧) ∪ 𝑡 ∈ dom 𝑧 suc (𝑧𝑡)) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
37 cfsmolem.2 . . . . . . . . . . . . . . . . . . . . . . 23 𝐹 = (𝑧 ∈ V ↦ ((𝑔‘dom 𝑧) ∪ 𝑡 ∈ dom 𝑧 suc (𝑧𝑡)))
38 fvex 6445 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∈ V
3929dmex 7360 . . . . . . . . . . . . . . . . . . . . . . . . 25 dom (recs(𝐹) ↾ 𝑥) ∈ V
40 fvex 6445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4140sucex 7271 . . . . . . . . . . . . . . . . . . . . . . . . 25 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4239, 41iunex 7407 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4338, 42unex 7215 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) ∈ V
4436, 37, 43fvmpt 6528 . . . . . . . . . . . . . . . . . . . . . 22 ((recs(𝐹) ↾ 𝑥) ∈ V → (𝐹‘(recs(𝐹) ↾ 𝑥)) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
4529, 44ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (𝐹‘(recs(𝐹) ↾ 𝑥)) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
4623, 45syl6eq 2876 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (recs(𝐹)‘𝑥) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
47 onss 7250 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ On → 𝑥 ⊆ On)
48 fnssres 6236 . . . . . . . . . . . . . . . . . . . . . 22 ((recs(𝐹) Fn On ∧ 𝑥 ⊆ On) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
4924, 47, 48sylancr 583 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
50 fndm 6222 . . . . . . . . . . . . . . . . . . . . 21 ((recs(𝐹) ↾ 𝑥) Fn 𝑥 → dom (recs(𝐹) ↾ 𝑥) = 𝑥)
51 fveq2 6432 . . . . . . . . . . . . . . . . . . . . . 22 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → (𝑔‘dom (recs(𝐹) ↾ 𝑥)) = (𝑔𝑥))
52 iuneq1 4753 . . . . . . . . . . . . . . . . . . . . . . 23 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
53 fvres 6451 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑡) = (recs(𝐹)‘𝑡))
54 suceq 6027 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((recs(𝐹) ↾ 𝑥)‘𝑡) = (recs(𝐹)‘𝑡) → suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = suc (recs(𝐹)‘𝑡))
5553, 54syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡𝑥 → suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = suc (recs(𝐹)‘𝑡))
5655iuneq2i 4758 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑡𝑥 suc (recs(𝐹)‘𝑡)
57 fveq2 6432 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑡 → (recs(𝐹)‘𝑦) = (recs(𝐹)‘𝑡))
58 suceq 6027 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((recs(𝐹)‘𝑦) = (recs(𝐹)‘𝑡) → suc (recs(𝐹)‘𝑦) = suc (recs(𝐹)‘𝑡))
5957, 58syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑡 → suc (recs(𝐹)‘𝑦) = suc (recs(𝐹)‘𝑡))
6059cbviunv 4778 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝑡𝑥 suc (recs(𝐹)‘𝑡)
6156, 60eqtr4i 2851 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑦𝑥 suc (recs(𝐹)‘𝑦)
6252, 61syl6eq 2876 . . . . . . . . . . . . . . . . . . . . . 22 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑦𝑥 suc (recs(𝐹)‘𝑦))
6351, 62uneq12d 3994 . . . . . . . . . . . . . . . . . . . . 21 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
6449, 50, 633syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
6546, 64eqtrd 2860 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → (recs(𝐹)‘𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
663, 65syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘𝐴) → (recs(𝐹)‘𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
6722, 66eqtrd 2860 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
68673ad2ant2 1170 . . . . . . . . . . . . . . . 16 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐺𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
69 eloni 5972 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On → Ord 𝐴)
7069adantl 475 . . . . . . . . . . . . . . . . . 18 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → Ord 𝐴)
71703ad2ant1 1169 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → Ord 𝐴)
72 f1f 6337 . . . . . . . . . . . . . . . . . . . 20 (𝑔:(cf‘𝐴)–1-1𝐴𝑔:(cf‘𝐴)⟶𝐴)
7372ffvelrnda 6607 . . . . . . . . . . . . . . . . . . 19 ((𝑔:(cf‘𝐴)–1-1𝐴𝑥 ∈ (cf‘𝐴)) → (𝑔𝑥) ∈ 𝐴)
7473adantlr 708 . . . . . . . . . . . . . . . . . 18 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (𝑔𝑥) ∈ 𝐴)
75743adant3 1168 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝑔𝑥) ∈ 𝐴)
7619fveq1i 6433 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺𝑦) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦)
7713fvresd 6452 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦) = (recs(𝐹)‘𝑦))
7876, 77syl5eq 2872 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
7978adantrl 709 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦𝑥 ∧ (𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴))) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
8079ancoms 452 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
8180eleq1d 2890 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 ↔ (recs(𝐹)‘𝑦) ∈ 𝐴))
82 ordsucss 7278 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝐴 → ((recs(𝐹)‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
8369, 82syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → ((recs(𝐹)‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
8483ad2antrr 719 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((recs(𝐹)‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
8581, 84sylbid 232 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
8685ralimdva 3170 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
87 iunss 4780 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 ↔ ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴)
8886, 87syl6ibr 244 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
89883impia 1151 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴)
90 onelon 5987 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ 𝐴) → (recs(𝐹)‘𝑦) ∈ On)
9190ex 403 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ On → ((recs(𝐹)‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
9291ad2antrr 719 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((recs(𝐹)‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
9381, 92sylbid 232 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
94 suceloni 7273 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((recs(𝐹)‘𝑦) ∈ On → suc (recs(𝐹)‘𝑦) ∈ On)
9593, 94syl6 35 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ∈ On))
9695ralimdva 3170 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On))
97963impia 1151 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
98 iunon 7701 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ V ∧ ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
9927, 98mpan 683 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
10097, 99syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
101 simp1 1172 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝐴 ∈ On)
102 onsseleq 6003 . . . . . . . . . . . . . . . . . . . . 21 (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On ∧ 𝐴 ∈ On) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 ↔ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴)))
103100, 101, 102syl2anc 581 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 ↔ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴)))
104 idd 24 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
105 simpll 785 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝑥 ∈ (cf‘𝐴))
106 simprr 791 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝐴 ∈ On)
1073ad2antrr 719 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝑥 ∈ On)
1083, 49syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (cf‘𝐴) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
109108adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
11078ancoms 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
111 fvres 6451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
112111adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
113110, 112eqtr4d 2863 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → (𝐺𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
114113eleq1d 2890 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 ↔ ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
115114ralbidva 3193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (cf‘𝐴) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 ↔ ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
116115biimpa 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
117 ffnfv 6636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((recs(𝐹) ↾ 𝑥):𝑥𝐴 ↔ ((recs(𝐹) ↾ 𝑥) Fn 𝑥 ∧ ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
118109, 116, 117sylanbrc 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥):𝑥𝐴)
119 eleq2 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 → (𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦) ↔ 𝑡𝐴))
120119biimpar 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝑡𝐴) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
121120adantrl 709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
1221213adant1 1166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
123 onelon 5987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝐴 ∈ On ∧ 𝑡𝐴) → 𝑡 ∈ On)
124111adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
125 ffvelrn 6605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
126124, 125eqeltrrd 2906 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → (recs(𝐹)‘𝑦) ∈ 𝐴)
127126, 90sylan2 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝐴 ∈ On ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
128127adantlr 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
129 onsssuc 6049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑡 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ On) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
130123, 128, 129syl2an2r 677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
131130anassrs 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) ∧ 𝑦𝑥) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
132131rexbidva 3258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
133 eliun 4743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦))
134132, 133syl6bbr 281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
135134ancoms 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
1361353adant2 1167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
137122, 136mpbird 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
1381373expa 1153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
139138anassrs 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) ∧ 𝑡𝐴) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
140139ralrimiva 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
141140expl 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((recs(𝐹) ↾ 𝑥):𝑥𝐴 → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
142118, 141syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
143142imp 397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
144 feq1 6258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓:𝑥𝐴 ↔ (recs(𝐹) ↾ 𝑥):𝑥𝐴))
145 fveq1 6431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
146145sseq2d 3857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑡 ⊆ (𝑓𝑦) ↔ 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
147146rexbidv 3261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
148111sseq2d 3857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦𝑥 → (𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ 𝑡 ⊆ (recs(𝐹)‘𝑦)))
149148rexbiia 3249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∃𝑦𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
150147, 149syl6bb 279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
151150ralbidv 3194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
152144, 151anbi12d 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 = (recs(𝐹) ↾ 𝑥) → ((𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)) ↔ ((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))))
15329, 152spcev 3516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)) → ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)))
154118, 143, 153syl2an2r 677 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)))
155 cfflb 9395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)) → (cf‘𝐴) ⊆ 𝑥))
156155imp 397 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦))) → (cf‘𝐴) ⊆ 𝑥)
157106, 107, 154, 156syl21anc 873 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → (cf‘𝐴) ⊆ 𝑥)
158 ontri1 5996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((cf‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
1592, 3, 158sylancr 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ (cf‘𝐴) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
160159ad2antrr 719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
161157, 160mpbid 224 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ¬ 𝑥 ∈ (cf‘𝐴))
162105, 161pm2.21dd 187 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
163162ex 403 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
164163expcomd 408 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐴 ∈ On → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)))
165164com12 32 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ On → ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)))
1661653impib 1150 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
167104, 166jaod 892 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
168103, 167sylbid 232 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
16989, 168mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
1701693adant1l 1227 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
171 ordunel 7287 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴 ∧ (𝑔𝑥) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴) → ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
17271, 75, 170, 171syl3anc 1496 . . . . . . . . . . . . . . . 16 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
17368, 172eqeltrd 2905 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐺𝑥) ∈ 𝐴)
1741733expia 1156 . . . . . . . . . . . . . 14 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → (𝐺𝑥) ∈ 𝐴))
1751743impa 1142 . . . . . . . . . . . . 13 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → (𝐺𝑥) ∈ 𝐴))
17618, 175syldc 48 . . . . . . . . . . . 12 (∀𝑦𝑥 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ 𝐴) → ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) ∈ 𝐴))
177176a1i 11 . . . . . . . . . . 11 (𝑥 ∈ On → (∀𝑦𝑥 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ 𝐴) → ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) ∈ 𝐴)))
1789, 177tfis2 7316 . . . . . . . . . 10 (𝑥 ∈ On → ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) ∈ 𝐴))
1794, 178mpcom 38 . . . . . . . . 9 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) ∈ 𝐴)
1801793expia 1156 . . . . . . . 8 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) ∈ 𝐴))
181180ralrimiv 3173 . . . . . . 7 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴)
1822onssi 7297 . . . . . . . . 9 (cf‘𝐴) ⊆ On
183 fnssres 6236 . . . . . . . . . 10 ((recs(𝐹) Fn On ∧ (cf‘𝐴) ⊆ On) → (recs(𝐹) ↾ (cf‘𝐴)) Fn (cf‘𝐴))
18419fneq1i 6217 . . . . . . . . . 10 (𝐺 Fn (cf‘𝐴) ↔ (recs(𝐹) ↾ (cf‘𝐴)) Fn (cf‘𝐴))
185183, 184sylibr 226 . . . . . . . . 9 ((recs(𝐹) Fn On ∧ (cf‘𝐴) ⊆ On) → 𝐺 Fn (cf‘𝐴))
18624, 182, 185mp2an 685 . . . . . . . 8 𝐺 Fn (cf‘𝐴)
187 ffnfv 6636 . . . . . . . 8 (𝐺:(cf‘𝐴)⟶𝐴 ↔ (𝐺 Fn (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴))
188186, 187mpbiran 702 . . . . . . 7 (𝐺:(cf‘𝐴)⟶𝐴 ↔ ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴)
189181, 188sylibr 226 . . . . . 6 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → 𝐺:(cf‘𝐴)⟶𝐴)
190189adantlr 708 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → 𝐺:(cf‘𝐴)⟶𝐴)
191 onss 7250 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
192191adantl 475 . . . . . . 7 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → 𝐴 ⊆ On)
1932onordi 6066 . . . . . . . 8 Ord (cf‘𝐴)
194 fvex 6445 . . . . . . . . . . . . . . . . 17 (recs(𝐹)‘𝑦) ∈ V
195194sucid 6041 . . . . . . . . . . . . . . . 16 (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)
196 fveq2 6432 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑦 → (recs(𝐹)‘𝑡) = (recs(𝐹)‘𝑦))
197 suceq 6027 . . . . . . . . . . . . . . . . . . 19 ((recs(𝐹)‘𝑡) = (recs(𝐹)‘𝑦) → suc (recs(𝐹)‘𝑡) = suc (recs(𝐹)‘𝑦))
198196, 197syl 17 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑦 → suc (recs(𝐹)‘𝑡) = suc (recs(𝐹)‘𝑦))
199198eliuni 4745 . . . . . . . . . . . . . . . . 17 ((𝑦𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ 𝑡𝑥 suc (recs(𝐹)‘𝑡))
200199, 60syl6eleqr 2916 . . . . . . . . . . . . . . . 16 ((𝑦𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
201195, 200mpan2 684 . . . . . . . . . . . . . . 15 (𝑦𝑥 → (recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
202 elun2 4007 . . . . . . . . . . . . . . 15 ((recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦) → (recs(𝐹)‘𝑦) ∈ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
203201, 202syl 17 . . . . . . . . . . . . . 14 (𝑦𝑥 → (recs(𝐹)‘𝑦) ∈ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
204203adantr 474 . . . . . . . . . . . . 13 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
2053adantl 475 . . . . . . . . . . . . . 14 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
206205, 65syl 17 . . . . . . . . . . . . 13 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
207204, 206eleqtrrd 2908 . . . . . . . . . . . 12 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ (recs(𝐹)‘𝑥))
20822adantl 475 . . . . . . . . . . . 12 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) = (recs(𝐹)‘𝑥))
209207, 78, 2083eltr4d 2920 . . . . . . . . . . 11 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ (𝐺𝑥))
210209expcom 404 . . . . . . . . . 10 (𝑥 ∈ (cf‘𝐴) → (𝑦𝑥 → (𝐺𝑦) ∈ (𝐺𝑥)))
211210ralrimiv 3173 . . . . . . . . 9 (𝑥 ∈ (cf‘𝐴) → ∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥))
212211rgen 3130 . . . . . . . 8 𝑥 ∈ (cf‘𝐴)∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥)
213 issmo2 7711 . . . . . . . . 9 (𝐺:(cf‘𝐴)⟶𝐴 → ((𝐴 ⊆ On ∧ Ord (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥)) → Smo 𝐺))
214213com12 32 . . . . . . . 8 ((𝐴 ⊆ On ∧ Ord (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥)) → (𝐺:(cf‘𝐴)⟶𝐴 → Smo 𝐺))
215193, 212, 214mp3an23 1583 . . . . . . 7 (𝐴 ⊆ On → (𝐺:(cf‘𝐴)⟶𝐴 → Smo 𝐺))
216192, 189, 215sylc 65 . . . . . 6 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → Smo 𝐺)
217216adantlr 708 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → Smo 𝐺)
218 fveq2 6432 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑔𝑥) = (𝑔𝑤))
219 fveq2 6432 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺𝑥) = (𝐺𝑤))
220218, 219sseq12d 3858 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑔𝑥) ⊆ (𝐺𝑥) ↔ (𝑔𝑤) ⊆ (𝐺𝑤)))
221 ssun1 4002 . . . . . . . . . . 11 (𝑔𝑥) ⊆ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
222221, 67syl5sseqr 3878 . . . . . . . . . 10 (𝑥 ∈ (cf‘𝐴) → (𝑔𝑥) ⊆ (𝐺𝑥))
223220, 222vtoclga 3488 . . . . . . . . 9 (𝑤 ∈ (cf‘𝐴) → (𝑔𝑤) ⊆ (𝐺𝑤))
224 sstr 3834 . . . . . . . . . 10 ((𝑧 ⊆ (𝑔𝑤) ∧ (𝑔𝑤) ⊆ (𝐺𝑤)) → 𝑧 ⊆ (𝐺𝑤))
225224expcom 404 . . . . . . . . 9 ((𝑔𝑤) ⊆ (𝐺𝑤) → (𝑧 ⊆ (𝑔𝑤) → 𝑧 ⊆ (𝐺𝑤)))
226223, 225syl 17 . . . . . . . 8 (𝑤 ∈ (cf‘𝐴) → (𝑧 ⊆ (𝑔𝑤) → 𝑧 ⊆ (𝐺𝑤)))
227226reximia 3216 . . . . . . 7 (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤) → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
228227ralimi 3160 . . . . . 6 (∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
229228ad2antlr 720 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
230 fnex 6736 . . . . . . 7 ((𝐺 Fn (cf‘𝐴) ∧ (cf‘𝐴) ∈ On) → 𝐺 ∈ V)
231186, 2, 230mp2an 685 . . . . . 6 𝐺 ∈ V
232 feq1 6258 . . . . . . 7 (𝑓 = 𝐺 → (𝑓:(cf‘𝐴)⟶𝐴𝐺:(cf‘𝐴)⟶𝐴))
233 smoeq 7712 . . . . . . 7 (𝑓 = 𝐺 → (Smo 𝑓 ↔ Smo 𝐺))
234 fveq1 6431 . . . . . . . . . 10 (𝑓 = 𝐺 → (𝑓𝑤) = (𝐺𝑤))
235234sseq2d 3857 . . . . . . . . 9 (𝑓 = 𝐺 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝐺𝑤)))
236235rexbidv 3261 . . . . . . . 8 (𝑓 = 𝐺 → (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤) ↔ ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)))
237236ralbidv 3194 . . . . . . 7 (𝑓 = 𝐺 → (∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤) ↔ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)))
238232, 233, 2373anbi123d 1566 . . . . . 6 (𝑓 = 𝐺 → ((𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)) ↔ (𝐺:(cf‘𝐴)⟶𝐴 ∧ Smo 𝐺 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))))
239231, 238spcev 3516 . . . . 5 ((𝐺:(cf‘𝐴)⟶𝐴 ∧ Smo 𝐺 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
240190, 217, 229, 239syl3anc 1496 . . . 4 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
241240expcom 404 . . 3 (𝐴 ∈ On → ((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
242241exlimdv 2034 . 2 (𝐴 ∈ On → (∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
2431, 242mpd 15 1 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 880  w3a 1113   = wceq 1658  wex 1880  wcel 2166  wral 3116  wrex 3117  Vcvv 3413  cun 3795  wss 3797   ciun 4739  cmpt 4951  dom cdm 5341  cres 5343  Ord word 5961  Oncon0 5962  suc csuc 5964  Fun wfun 6116   Fn wfn 6117  wf 6118  1-1wf1 6119  cfv 6122  Smo wsmo 7707  recscrecs 7732  cfccf 9075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-se 5301  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-isom 6131  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-smo 7708  df-recs 7733  df-er 8008  df-map 8123  df-en 8222  df-dom 8223  df-sdom 8224  df-card 9077  df-cf 9079  df-acn 9080
This theorem is referenced by:  cfsmo  9407
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