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Theorem cfsmolem 10267
Description: Lemma for cfsmo 10268. (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 10255 . 2 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)))
2 cfon 10252 . . . . . . . . . . . 12 (cf‘𝐴) ∈ On
32oneli 6477 . . . . . . . . . . 11 (𝑥 ∈ (cf‘𝐴) → 𝑥 ∈ On)
433ad2ant3 1133 . . . . . . . . . 10 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
5 eleq1w 2814 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥 ∈ (cf‘𝐴) ↔ 𝑦 ∈ (cf‘𝐴)))
653anbi3d 1440 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ↔ (𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴))))
7 fveq2 6890 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
87eleq1d 2816 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐺𝑥) ∈ 𝐴 ↔ (𝐺𝑦) ∈ 𝐴))
96, 8imbi12d 343 . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → 𝑔:(cf‘𝐴)–1-1𝐴)
11 simpl2 1190 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → 𝐴 ∈ On)
12 ontr1 6409 . . . . . . . . . . . . . . . . . 18 ((cf‘𝐴) ∈ On → ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴)))
132, 12ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴))
1413ancoms 457 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → 𝑦 ∈ (cf‘𝐴))
15143ad2antl3 1185 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . . . 19 (𝐺𝑥) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥)
21 fvres 6909 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (cf‘𝐴) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥) = (recs(𝐹)‘𝑥))
2220, 21eqtrid 2782 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) = (recs(𝐹)‘𝑥))
23 recsval 8406 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → (recs(𝐹)‘𝑥) = (𝐹‘(recs(𝐹) ↾ 𝑥)))
24 recsfnon 8405 . . . . . . . . . . . . . . . . . . . . . . . 24 recs(𝐹) Fn On
25 fnfun 6648 . . . . . . . . . . . . . . . . . . . . . . . 24 (recs(𝐹) Fn On → Fun recs(𝐹))
2624, 25ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 Fun recs(𝐹)
27 vex 3476 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥 ∈ V
28 resfunexg 7218 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun recs(𝐹) ∧ 𝑥 ∈ V) → (recs(𝐹) ↾ 𝑥) ∈ V)
2926, 27, 28mp2an 688 . . . . . . . . . . . . . . . . . . . . . 22 (recs(𝐹) ↾ 𝑥) ∈ V
30 dmeq 5902 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (recs(𝐹) ↾ 𝑥) → dom 𝑧 = dom (recs(𝐹) ↾ 𝑥))
3130fveq2d 6894 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑔‘dom 𝑧) = (𝑔‘dom (recs(𝐹) ↾ 𝑥)))
32 fveq1 6889 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑧𝑡) = ((recs(𝐹) ↾ 𝑥)‘𝑡))
33 suceq 6429 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5024 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (recs(𝐹) ↾ 𝑥) → 𝑡 ∈ dom 𝑧 suc (𝑧𝑡) = 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
3631, 35uneq12d 4163 . . . . . . . . . . . . . . . . . . . . . . 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 6903 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∈ V
3929dmex 7904 . . . . . . . . . . . . . . . . . . . . . . . . 25 dom (recs(𝐹) ↾ 𝑥) ∈ V
40 fvex 6903 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4140sucex 7796 . . . . . . . . . . . . . . . . . . . . . . . . 25 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4239, 41iunex 7957 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4338, 42unex 7735 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) ∈ V
4436, 37, 43fvmpt 6997 . . . . . . . . . . . . . . . . . . . . . 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, 45eqtrdi 2786 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (recs(𝐹)‘𝑥) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
47 onss 7774 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ On → 𝑥 ⊆ On)
48 fnssres 6672 . . . . . . . . . . . . . . . . . . . . . 22 ((recs(𝐹) Fn On ∧ 𝑥 ⊆ On) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
4924, 47, 48sylancr 585 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
50 fndm 6651 . . . . . . . . . . . . . . . . . . . . 21 ((recs(𝐹) ↾ 𝑥) Fn 𝑥 → dom (recs(𝐹) ↾ 𝑥) = 𝑥)
51 fveq2 6890 . . . . . . . . . . . . . . . . . . . . . 22 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → (𝑔‘dom (recs(𝐹) ↾ 𝑥)) = (𝑔𝑥))
52 iuneq1 5012 . . . . . . . . . . . . . . . . . . . . . . 23 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
53 fvres 6909 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑡) = (recs(𝐹)‘𝑡))
54 suceq 6429 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5017 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑡𝑥 suc (recs(𝐹)‘𝑡)
57 fveq2 6890 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑡 → (recs(𝐹)‘𝑦) = (recs(𝐹)‘𝑡))
58 suceq 6429 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5042 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝑡𝑥 suc (recs(𝐹)‘𝑡)
6156, 60eqtr4i 2761 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑦𝑥 suc (recs(𝐹)‘𝑦)
6252, 61eqtrdi 2786 . . . . . . . . . . . . . . . . . . . . . 22 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑦𝑥 suc (recs(𝐹)‘𝑦))
6351, 62uneq12d 4163 . . . . . . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
68673ad2ant2 1132 . . . . . . . . . . . . . . . 16 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐺𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
69 eloni 6373 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On → Ord 𝐴)
7069adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → Ord 𝐴)
71703ad2ant1 1131 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → Ord 𝐴)
72 f1f 6786 . . . . . . . . . . . . . . . . . . . 20 (𝑔:(cf‘𝐴)–1-1𝐴𝑔:(cf‘𝐴)⟶𝐴)
7372ffvelcdmda 7085 . . . . . . . . . . . . . . . . . . 19 ((𝑔:(cf‘𝐴)–1-1𝐴𝑥 ∈ (cf‘𝐴)) → (𝑔𝑥) ∈ 𝐴)
7473adantlr 711 . . . . . . . . . . . . . . . . . 18 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (𝑔𝑥) ∈ 𝐴)
75743adant3 1130 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝑔𝑥) ∈ 𝐴)
7619fveq1i 6891 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺𝑦) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦)
7713fvresd 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦) = (recs(𝐹)‘𝑦))
7876, 77eqtrid 2782 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
7978adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦𝑥 ∧ (𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴))) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
8079ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
8180eleq1d 2816 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 ↔ (recs(𝐹)‘𝑦) ∈ 𝐴))
82 ordsucss 7808 . . . . . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((recs(𝐹)‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
8581, 84sylbid 239 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
8685ralimdva 3165 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
87 iunss 5047 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 ↔ ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴)
8886, 87imbitrrdi 251 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
89883impia 1115 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴)
90 onelon 6388 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ 𝐴) → (recs(𝐹)‘𝑦) ∈ On)
9190ex 411 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ On → ((recs(𝐹)‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
9291ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((recs(𝐹)‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
9381, 92sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
94 onsuc 7801 . . . . . . . . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On))
97963impia 1115 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
98 iunon 8341 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ V ∧ ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
9927, 97, 98sylancr 585 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
100 simp1 1134 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝐴 ∈ On)
101 onsseleq 6404 . . . . . . . . . . . . . . . . . . . . 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)))
10299, 100, 101syl2anc 582 . . . . . . . . . . . . . . . . . . . 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)))
103 idd 24 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
104 simpll 763 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝑥 ∈ (cf‘𝐴))
105 simprr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝐴 ∈ On)
1063ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝑥 ∈ On)
1073, 49syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (cf‘𝐴) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
108107adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
10978ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
110 fvres 6909 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
111110adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
112109, 111eqtr4d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → (𝐺𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
113112eleq1d 2816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 ↔ ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
114113ralbidva 3173 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (cf‘𝐴) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 ↔ ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
115114biimpa 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
116 ffnfv 7119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((recs(𝐹) ↾ 𝑥):𝑥𝐴 ↔ ((recs(𝐹) ↾ 𝑥) Fn 𝑥 ∧ ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
117108, 115, 116sylanbrc 581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥):𝑥𝐴)
118 eleq2 2820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 → (𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦) ↔ 𝑡𝐴))
119118biimpar 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝑡𝐴) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
120119adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
1211203adant1 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
122 onelon 6388 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝐴 ∈ On ∧ 𝑡𝐴) → 𝑡 ∈ On)
123110adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
124 ffvelcdm 7082 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
125123, 124eqeltrrd 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → (recs(𝐹)‘𝑦) ∈ 𝐴)
126125, 90sylan2 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝐴 ∈ On ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
127126adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
128 onsssuc 6453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑡 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ On) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
129122, 127, 128syl2an2r 681 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
130129anassrs 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) ∧ 𝑦𝑥) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
131130rexbidva 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
132 eliun 5000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦))
133131, 132bitr4di 288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
134133ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
1351343adant2 1129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
136121, 135mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
1371363expa 1116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
138137anassrs 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) ∧ 𝑡𝐴) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
139138ralrimiva 3144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
140139expl 456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((recs(𝐹) ↾ 𝑥):𝑥𝐴 → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
141117, 140syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
142141imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
143 feq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓:𝑥𝐴 ↔ (recs(𝐹) ↾ 𝑥):𝑥𝐴))
144 fveq1 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
145144sseq2d 4013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑡 ⊆ (𝑓𝑦) ↔ 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
146145rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
147110sseq2d 4013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦𝑥 → (𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ 𝑡 ⊆ (recs(𝐹)‘𝑦)))
148147rexbiia 3090 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∃𝑦𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
149146, 148bitrdi 286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
150149ralbidv 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
151143, 150anbi12d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 = (recs(𝐹) ↾ 𝑥) → ((𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)) ↔ ((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))))
15229, 151spcev 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)) → ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)))
153117, 142, 152syl2an2r 681 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)))
154 cfflb 10256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)) → (cf‘𝐴) ⊆ 𝑥))
155154imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦))) → (cf‘𝐴) ⊆ 𝑥)
156105, 106, 153, 155syl21anc 834 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → (cf‘𝐴) ⊆ 𝑥)
157 ontri1 6397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((cf‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
1582, 3, 157sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ (cf‘𝐴) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
159158ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
160156, 159mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ¬ 𝑥 ∈ (cf‘𝐴))
161104, 160pm2.21dd 194 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
162161ex 411 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
163162expcomd 415 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐴 ∈ On → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)))
164163com12 32 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ On → ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)))
1651643impib 1114 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
166103, 165jaod 855 . . . . . . . . . . . . . . . . . . . 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))
167102, 166sylbid 239 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
16889, 167mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
1691683adant1l 1174 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
170 ordunel 7817 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴 ∧ (𝑔𝑥) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴) → ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
17171, 75, 169, 170syl3anc 1369 . . . . . . . . . . . . . . . 16 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
17268, 171eqeltrd 2831 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐺𝑥) ∈ 𝐴)
1731723expia 1119 . . . . . . . . . . . . . 14 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → (𝐺𝑥) ∈ 𝐴))
1741733impa 1108 . . . . . . . . . . . . 13 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → (𝐺𝑥) ∈ 𝐴))
17518, 174syldc 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))
176175a1i 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)))
1779, 176tfis2 7848 . . . . . . . . . 10 (𝑥 ∈ On → ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) ∈ 𝐴))
1784, 177mpcom 38 . . . . . . . . 9 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) ∈ 𝐴)
1791783expia 1119 . . . . . . . 8 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) ∈ 𝐴))
180179ralrimiv 3143 . . . . . . 7 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴)
1812onssi 7828 . . . . . . . . 9 (cf‘𝐴) ⊆ On
182 fnssres 6672 . . . . . . . . . 10 ((recs(𝐹) Fn On ∧ (cf‘𝐴) ⊆ On) → (recs(𝐹) ↾ (cf‘𝐴)) Fn (cf‘𝐴))
18319fneq1i 6645 . . . . . . . . . 10 (𝐺 Fn (cf‘𝐴) ↔ (recs(𝐹) ↾ (cf‘𝐴)) Fn (cf‘𝐴))
184182, 183sylibr 233 . . . . . . . . 9 ((recs(𝐹) Fn On ∧ (cf‘𝐴) ⊆ On) → 𝐺 Fn (cf‘𝐴))
18524, 181, 184mp2an 688 . . . . . . . 8 𝐺 Fn (cf‘𝐴)
186 ffnfv 7119 . . . . . . . 8 (𝐺:(cf‘𝐴)⟶𝐴 ↔ (𝐺 Fn (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴))
187185, 186mpbiran 705 . . . . . . 7 (𝐺:(cf‘𝐴)⟶𝐴 ↔ ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴)
188180, 187sylibr 233 . . . . . 6 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → 𝐺:(cf‘𝐴)⟶𝐴)
189188adantlr 711 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → 𝐺:(cf‘𝐴)⟶𝐴)
190 onss 7774 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
191190adantl 480 . . . . . . 7 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → 𝐴 ⊆ On)
1922onordi 6474 . . . . . . . 8 Ord (cf‘𝐴)
193 fvex 6903 . . . . . . . . . . . . . . . . 17 (recs(𝐹)‘𝑦) ∈ V
194193sucid 6445 . . . . . . . . . . . . . . . 16 (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)
195 fveq2 6890 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑦 → (recs(𝐹)‘𝑡) = (recs(𝐹)‘𝑦))
196 suceq 6429 . . . . . . . . . . . . . . . . . . 19 ((recs(𝐹)‘𝑡) = (recs(𝐹)‘𝑦) → suc (recs(𝐹)‘𝑡) = suc (recs(𝐹)‘𝑦))
197195, 196syl 17 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑦 → suc (recs(𝐹)‘𝑡) = suc (recs(𝐹)‘𝑦))
198197eliuni 5002 . . . . . . . . . . . . . . . . 17 ((𝑦𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ 𝑡𝑥 suc (recs(𝐹)‘𝑡))
199198, 60eleqtrrdi 2842 . . . . . . . . . . . . . . . 16 ((𝑦𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
200194, 199mpan2 687 . . . . . . . . . . . . . . 15 (𝑦𝑥 → (recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
201 elun2 4176 . . . . . . . . . . . . . . 15 ((recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦) → (recs(𝐹)‘𝑦) ∈ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
202200, 201syl 17 . . . . . . . . . . . . . 14 (𝑦𝑥 → (recs(𝐹)‘𝑦) ∈ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
203202adantr 479 . . . . . . . . . . . . 13 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
2043adantl 480 . . . . . . . . . . . . . 14 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
205204, 65syl 17 . . . . . . . . . . . . 13 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
206203, 205eleqtrrd 2834 . . . . . . . . . . . 12 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ (recs(𝐹)‘𝑥))
20722adantl 480 . . . . . . . . . . . 12 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) = (recs(𝐹)‘𝑥))
208206, 78, 2073eltr4d 2846 . . . . . . . . . . 11 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ (𝐺𝑥))
209208expcom 412 . . . . . . . . . 10 (𝑥 ∈ (cf‘𝐴) → (𝑦𝑥 → (𝐺𝑦) ∈ (𝐺𝑥)))
210209ralrimiv 3143 . . . . . . . . 9 (𝑥 ∈ (cf‘𝐴) → ∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥))
211210rgen 3061 . . . . . . . 8 𝑥 ∈ (cf‘𝐴)∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥)
212 issmo2 8351 . . . . . . . . 9 (𝐺:(cf‘𝐴)⟶𝐴 → ((𝐴 ⊆ On ∧ Ord (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥)) → Smo 𝐺))
213212com12 32 . . . . . . . 8 ((𝐴 ⊆ On ∧ Ord (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥)) → (𝐺:(cf‘𝐴)⟶𝐴 → Smo 𝐺))
214192, 211, 213mp3an23 1451 . . . . . . 7 (𝐴 ⊆ On → (𝐺:(cf‘𝐴)⟶𝐴 → Smo 𝐺))
215191, 188, 214sylc 65 . . . . . 6 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → Smo 𝐺)
216215adantlr 711 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → Smo 𝐺)
217 fveq2 6890 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑔𝑥) = (𝑔𝑤))
218 fveq2 6890 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺𝑥) = (𝐺𝑤))
219217, 218sseq12d 4014 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑔𝑥) ⊆ (𝐺𝑥) ↔ (𝑔𝑤) ⊆ (𝐺𝑤)))
220 ssun1 4171 . . . . . . . . . . 11 (𝑔𝑥) ⊆ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
221220, 67sseqtrrid 4034 . . . . . . . . . 10 (𝑥 ∈ (cf‘𝐴) → (𝑔𝑥) ⊆ (𝐺𝑥))
222219, 221vtoclga 3565 . . . . . . . . 9 (𝑤 ∈ (cf‘𝐴) → (𝑔𝑤) ⊆ (𝐺𝑤))
223 sstr 3989 . . . . . . . . . 10 ((𝑧 ⊆ (𝑔𝑤) ∧ (𝑔𝑤) ⊆ (𝐺𝑤)) → 𝑧 ⊆ (𝐺𝑤))
224223expcom 412 . . . . . . . . 9 ((𝑔𝑤) ⊆ (𝐺𝑤) → (𝑧 ⊆ (𝑔𝑤) → 𝑧 ⊆ (𝐺𝑤)))
225222, 224syl 17 . . . . . . . 8 (𝑤 ∈ (cf‘𝐴) → (𝑧 ⊆ (𝑔𝑤) → 𝑧 ⊆ (𝐺𝑤)))
226225reximia 3079 . . . . . . 7 (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤) → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
227226ralimi 3081 . . . . . 6 (∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
228227ad2antlr 723 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
229 fnex 7220 . . . . . . 7 ((𝐺 Fn (cf‘𝐴) ∧ (cf‘𝐴) ∈ On) → 𝐺 ∈ V)
230185, 2, 229mp2an 688 . . . . . 6 𝐺 ∈ V
231 feq1 6697 . . . . . . 7 (𝑓 = 𝐺 → (𝑓:(cf‘𝐴)⟶𝐴𝐺:(cf‘𝐴)⟶𝐴))
232 smoeq 8352 . . . . . . 7 (𝑓 = 𝐺 → (Smo 𝑓 ↔ Smo 𝐺))
233 fveq1 6889 . . . . . . . . . 10 (𝑓 = 𝐺 → (𝑓𝑤) = (𝐺𝑤))
234233sseq2d 4013 . . . . . . . . 9 (𝑓 = 𝐺 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝐺𝑤)))
235234rexbidv 3176 . . . . . . . 8 (𝑓 = 𝐺 → (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤) ↔ ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)))
236235ralbidv 3175 . . . . . . 7 (𝑓 = 𝐺 → (∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤) ↔ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)))
237231, 232, 2363anbi123d 1434 . . . . . 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))))
238230, 237spcev 3595 . . . . 5 ((𝐺:(cf‘𝐴)⟶𝐴 ∧ Smo 𝐺 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
239189, 216, 228, 238syl3anc 1369 . . . 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)))
240239expcom 412 . . 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))))
241240exlimdv 1934 . 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))))
2421, 241mpd 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 205  wa 394  wo 843  w3a 1085   = wceq 1539  wex 1779  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  cun 3945  wss 3947   ciun 4996  cmpt 5230  dom cdm 5675  cres 5677  Ord word 6362  Oncon0 6363  suc csuc 6365  Fun wfun 6536   Fn wfn 6537  wf 6538  1-1wf1 6539  cfv 6542  Smo wsmo 8347  recscrecs 8372  cfccf 9934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-smo 8348  df-recs 8373  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-card 9936  df-cf 9938  df-acn 9939
This theorem is referenced by:  cfsmo  10268
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