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Theorem cfsmolem 10269
Description: Lemma for cfsmo 10270. (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 10257 . 2 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)))
2 cfon 10254 . . . . . . . . . . . 12 (cf‘𝐴) ∈ On
32oneli 6478 . . . . . . . . . . 11 (𝑥 ∈ (cf‘𝐴) → 𝑥 ∈ On)
433ad2ant3 1134 . . . . . . . . . 10 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
5 eleq1w 2815 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥 ∈ (cf‘𝐴) ↔ 𝑦 ∈ (cf‘𝐴)))
653anbi3d 1441 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ↔ (𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴))))
7 fveq2 6891 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
87eleq1d 2817 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐺𝑥) ∈ 𝐴 ↔ (𝐺𝑦) ∈ 𝐴))
96, 8imbi12d 344 . . . . . . . . . . 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 1190 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → 𝑔:(cf‘𝐴)–1-1𝐴)
11 simpl2 1191 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → 𝐴 ∈ On)
12 ontr1 6410 . . . . . . . . . . . . . . . . . 18 ((cf‘𝐴) ∈ On → ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴)))
132, 12ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴))
1413ancoms 458 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → 𝑦 ∈ (cf‘𝐴))
15143ad2antl3 1186 . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . 19 (𝐺𝑥) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥)
21 fvres 6910 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (cf‘𝐴) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥) = (recs(𝐹)‘𝑥))
2220, 21eqtrid 2783 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) = (recs(𝐹)‘𝑥))
23 recsval 8408 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → (recs(𝐹)‘𝑥) = (𝐹‘(recs(𝐹) ↾ 𝑥)))
24 recsfnon 8407 . . . . . . . . . . . . . . . . . . . . . . . 24 recs(𝐹) Fn On
25 fnfun 6649 . . . . . . . . . . . . . . . . . . . . . . . 24 (recs(𝐹) Fn On → Fun recs(𝐹))
2624, 25ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 Fun recs(𝐹)
27 vex 3477 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥 ∈ V
28 resfunexg 7219 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun recs(𝐹) ∧ 𝑥 ∈ V) → (recs(𝐹) ↾ 𝑥) ∈ V)
2926, 27, 28mp2an 689 . . . . . . . . . . . . . . . . . . . . . 22 (recs(𝐹) ↾ 𝑥) ∈ V
30 dmeq 5903 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (recs(𝐹) ↾ 𝑥) → dom 𝑧 = dom (recs(𝐹) ↾ 𝑥))
3130fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑔‘dom 𝑧) = (𝑔‘dom (recs(𝐹) ↾ 𝑥)))
32 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑧𝑡) = ((recs(𝐹) ↾ 𝑥)‘𝑡))
33 suceq 6430 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5025 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (recs(𝐹) ↾ 𝑥) → 𝑡 ∈ dom 𝑧 suc (𝑧𝑡) = 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
3631, 35uneq12d 4164 . . . . . . . . . . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∈ V
3929dmex 7906 . . . . . . . . . . . . . . . . . . . . . . . . 25 dom (recs(𝐹) ↾ 𝑥) ∈ V
40 fvex 6904 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4140sucex 7798 . . . . . . . . . . . . . . . . . . . . . . . . 25 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4239, 41iunex 7959 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
4338, 42unex 7737 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) ∈ V
4436, 37, 43fvmpt 6998 . . . . . . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (recs(𝐹)‘𝑥) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
47 onss 7776 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ On → 𝑥 ⊆ On)
48 fnssres 6673 . . . . . . . . . . . . . . . . . . . . . 22 ((recs(𝐹) Fn On ∧ 𝑥 ⊆ On) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
4924, 47, 48sylancr 586 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
50 fndm 6652 . . . . . . . . . . . . . . . . . . . . 21 ((recs(𝐹) ↾ 𝑥) Fn 𝑥 → dom (recs(𝐹) ↾ 𝑥) = 𝑥)
51 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . 22 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → (𝑔‘dom (recs(𝐹) ↾ 𝑥)) = (𝑔𝑥))
52 iuneq1 5013 . . . . . . . . . . . . . . . . . . . . . . 23 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
53 fvres 6910 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑡) = (recs(𝐹)‘𝑡))
54 suceq 6430 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5018 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑡𝑥 suc (recs(𝐹)‘𝑡)
57 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑡 → (recs(𝐹)‘𝑦) = (recs(𝐹)‘𝑡))
58 suceq 6430 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5043 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝑡𝑥 suc (recs(𝐹)‘𝑡)
6156, 60eqtr4i 2762 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑦𝑥 suc (recs(𝐹)‘𝑦)
6252, 61eqtrdi 2787 . . . . . . . . . . . . . . . . . . . . . 22 (dom (recs(𝐹) ↾ 𝑥) = 𝑥 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = 𝑦𝑥 suc (recs(𝐹)‘𝑦))
6351, 62uneq12d 4164 . . . . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
68673ad2ant2 1133 . . . . . . . . . . . . . . . 16 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐺𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
69 eloni 6374 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On → Ord 𝐴)
7069adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → Ord 𝐴)
71703ad2ant1 1132 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → Ord 𝐴)
72 f1f 6787 . . . . . . . . . . . . . . . . . . . 20 (𝑔:(cf‘𝐴)–1-1𝐴𝑔:(cf‘𝐴)⟶𝐴)
7372ffvelcdmda 7086 . . . . . . . . . . . . . . . . . . 19 ((𝑔:(cf‘𝐴)–1-1𝐴𝑥 ∈ (cf‘𝐴)) → (𝑔𝑥) ∈ 𝐴)
7473adantlr 712 . . . . . . . . . . . . . . . . . 18 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (𝑔𝑥) ∈ 𝐴)
75743adant3 1131 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝑔𝑥) ∈ 𝐴)
7619fveq1i 6892 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺𝑦) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦)
7713fvresd 6911 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦) = (recs(𝐹)‘𝑦))
7876, 77eqtrid 2783 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
7978adantrl 713 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦𝑥 ∧ (𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴))) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
8079ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
8180eleq1d 2817 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 ↔ (recs(𝐹)‘𝑦) ∈ 𝐴))
82 ordsucss 7810 . . . . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
87 iunss 5048 . . . . . . . . . . . . . . . . . . . . 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 1116 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴)
90 onelon 6389 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ 𝐴) → (recs(𝐹)‘𝑦) ∈ On)
9190ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ On → ((recs(𝐹)‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
9291ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7803 . . . . . . . . . . . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On))
97963impia 1116 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
98 iunon 8343 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ V ∧ ∀𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
9927, 97, 98sylancr 586 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
100 simp1 1135 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝐴 ∈ On)
101 onsseleq 6405 . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝑥 ∈ (cf‘𝐴))
105 simprr 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → 𝐴 ∈ On)
1063ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
10978ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → (𝐺𝑦) = (recs(𝐹)‘𝑦))
110 fvres 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
111110adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
112109, 111eqtr4d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → (𝐺𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
113112eleq1d 2817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦𝑥) → ((𝐺𝑦) ∈ 𝐴 ↔ ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
114113ralbidva 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (cf‘𝐴) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 ↔ ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
115114biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
116 ffnfv 7120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((recs(𝐹) ↾ 𝑥):𝑥𝐴 ↔ ((recs(𝐹) ↾ 𝑥) Fn 𝑥 ∧ ∀𝑦𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
117108, 115, 116sylanbrc 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥):𝑥𝐴)
118 eleq2 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 → (𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦) ↔ 𝑡𝐴))
119118biimpar 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝑡𝐴) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
120119adantrl 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
1211203adant1 1129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦))
122 onelon 6389 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝐴 ∈ On ∧ 𝑡𝐴) → 𝑡 ∈ On)
123110adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
124 ffvelcdm 7083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
125123, 124eqeltrrd 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥) → (recs(𝐹)‘𝑦) ∈ 𝐴)
126125, 90sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝐴 ∈ On ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
127126adantlr 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
128 onsssuc 6454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑡 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ On) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
129122, 127, 128syl2an2r 682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥𝐴𝑦𝑥)) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
130129anassrs 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) ∧ 𝑦𝑥) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
131130rexbidva 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
132 eliun 5001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦))
133131, 132bitr4di 289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝐴 ∈ On ∧ 𝑡𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥𝐴) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
134133ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → (∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
1351343adant2 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
1371363expa 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ (𝐴 ∈ On ∧ 𝑡𝐴)) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
138137anassrs 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) ∧ 𝑡𝐴) → ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
139138ralrimiva 3145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((recs(𝐹) ↾ 𝑥):𝑥𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
140139expl 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
143 feq1 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓:𝑥𝐴 ↔ (recs(𝐹) ↾ 𝑥):𝑥𝐴))
144 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
145144sseq2d 4014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑡 ⊆ (𝑓𝑦) ↔ 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
146145rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
147110sseq2d 4014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦𝑥 → (𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ 𝑡 ⊆ (recs(𝐹)‘𝑦)))
148147rexbiia 3091 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∃𝑦𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
149146, 148bitrdi 287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∃𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
150149ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 = (recs(𝐹) ↾ 𝑥) → (∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦) ↔ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
151143, 150anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 = (recs(𝐹) ↾ 𝑥) → ((𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)) ↔ ((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))))
15229, 151spcev 3596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((recs(𝐹) ↾ 𝑥):𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)) → ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)))
153117, 142, 152syl2an2r 682 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)))
154 cfflb 10258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦)) → (cf‘𝐴) ⊆ 𝑥))
155154imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∃𝑓(𝑓:𝑥𝐴 ∧ ∀𝑡𝐴𝑦𝑥 𝑡 ⊆ (𝑓𝑦))) → (cf‘𝐴) ⊆ 𝑥)
156105, 106, 153, 155syl21anc 835 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) ∧ ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On)) → (cf‘𝐴) ⊆ 𝑥)
157 ontri1 6398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((cf‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
1582, 3, 157sylancr 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ (cf‘𝐴) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
159158ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴𝐴 ∈ On) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
163162expcomd 416 . . . . . . . . . . . . . . . . . . . . . . 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 1115 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ( 𝑦𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
166103, 165jaod 856 . . . . . . . . . . . . . . . . . . . 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 1175 . . . . . . . . . . . . . . . . 17 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
170 ordunel 7819 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴 ∧ (𝑔𝑥) ∈ 𝐴 𝑦𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴) → ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
17171, 75, 169, 170syl3anc 1370 . . . . . . . . . . . . . . . 16 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
17268, 171eqeltrd 2832 . . . . . . . . . . . . . . 15 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴) → (𝐺𝑥) ∈ 𝐴)
1731723expia 1120 . . . . . . . . . . . . . 14 (((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦𝑥 (𝐺𝑦) ∈ 𝐴 → (𝐺𝑥) ∈ 𝐴))
1741733impa 1109 . . . . . . . . . . . . 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 7850 . . . . . . . . . 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 1120 . . . . . . . 8 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → (𝑥 ∈ (cf‘𝐴) → (𝐺𝑥) ∈ 𝐴))
180179ralrimiv 3144 . . . . . . 7 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴)
1812onssi 7830 . . . . . . . . 9 (cf‘𝐴) ⊆ On
182 fnssres 6673 . . . . . . . . . 10 ((recs(𝐹) Fn On ∧ (cf‘𝐴) ⊆ On) → (recs(𝐹) ↾ (cf‘𝐴)) Fn (cf‘𝐴))
18319fneq1i 6646 . . . . . . . . . 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 689 . . . . . . . 8 𝐺 Fn (cf‘𝐴)
186 ffnfv 7120 . . . . . . . 8 (𝐺:(cf‘𝐴)⟶𝐴 ↔ (𝐺 Fn (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)(𝐺𝑥) ∈ 𝐴))
187185, 186mpbiran 706 . . . . . . 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 712 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → 𝐺:(cf‘𝐴)⟶𝐴)
190 onss 7776 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
191190adantl 481 . . . . . . 7 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → 𝐴 ⊆ On)
1922onordi 6475 . . . . . . . 8 Ord (cf‘𝐴)
193 fvex 6904 . . . . . . . . . . . . . . . . 17 (recs(𝐹)‘𝑦) ∈ V
194193sucid 6446 . . . . . . . . . . . . . . . 16 (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)
195 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑦 → (recs(𝐹)‘𝑡) = (recs(𝐹)‘𝑦))
196 suceq 6430 . . . . . . . . . . . . . . . . . . 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 5003 . . . . . . . . . . . . . . . . 17 ((𝑦𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ 𝑡𝑥 suc (recs(𝐹)‘𝑡))
199198, 60eleqtrrdi 2843 . . . . . . . . . . . . . . . 16 ((𝑦𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
200194, 199mpan2 688 . . . . . . . . . . . . . . 15 (𝑦𝑥 → (recs(𝐹)‘𝑦) ∈ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
201 elun2 4177 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
2043adantl 481 . . . . . . . . . . . . . 14 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
205204, 65syl 17 . . . . . . . . . . . . 13 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑥) = ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦)))
206203, 205eleqtrrd 2835 . . . . . . . . . . . 12 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ (recs(𝐹)‘𝑥))
20722adantl 481 . . . . . . . . . . . 12 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑥) = (recs(𝐹)‘𝑥))
208206, 78, 2073eltr4d 2847 . . . . . . . . . . 11 ((𝑦𝑥𝑥 ∈ (cf‘𝐴)) → (𝐺𝑦) ∈ (𝐺𝑥))
209208expcom 413 . . . . . . . . . 10 (𝑥 ∈ (cf‘𝐴) → (𝑦𝑥 → (𝐺𝑦) ∈ (𝐺𝑥)))
210209ralrimiv 3144 . . . . . . . . 9 (𝑥 ∈ (cf‘𝐴) → ∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥))
211210rgen 3062 . . . . . . . 8 𝑥 ∈ (cf‘𝐴)∀𝑦𝑥 (𝐺𝑦) ∈ (𝐺𝑥)
212 issmo2 8353 . . . . . . . . 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 1452 . . . . . . 7 (𝐴 ⊆ On → (𝐺:(cf‘𝐴)⟶𝐴 → Smo 𝐺))
215191, 188, 214sylc 65 . . . . . 6 ((𝑔:(cf‘𝐴)–1-1𝐴𝐴 ∈ On) → Smo 𝐺)
216215adantlr 712 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → Smo 𝐺)
217 fveq2 6891 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑔𝑥) = (𝑔𝑤))
218 fveq2 6891 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺𝑥) = (𝐺𝑤))
219217, 218sseq12d 4015 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑔𝑥) ⊆ (𝐺𝑥) ↔ (𝑔𝑤) ⊆ (𝐺𝑤)))
220 ssun1 4172 . . . . . . . . . . 11 (𝑔𝑥) ⊆ ((𝑔𝑥) ∪ 𝑦𝑥 suc (recs(𝐹)‘𝑦))
221220, 67sseqtrrid 4035 . . . . . . . . . 10 (𝑥 ∈ (cf‘𝐴) → (𝑔𝑥) ⊆ (𝐺𝑥))
222219, 221vtoclga 3566 . . . . . . . . 9 (𝑤 ∈ (cf‘𝐴) → (𝑔𝑤) ⊆ (𝐺𝑤))
223 sstr 3990 . . . . . . . . . 10 ((𝑧 ⊆ (𝑔𝑤) ∧ (𝑔𝑤) ⊆ (𝐺𝑤)) → 𝑧 ⊆ (𝐺𝑤))
224223expcom 413 . . . . . . . . 9 ((𝑔𝑤) ⊆ (𝐺𝑤) → (𝑧 ⊆ (𝑔𝑤) → 𝑧 ⊆ (𝐺𝑤)))
225222, 224syl 17 . . . . . . . 8 (𝑤 ∈ (cf‘𝐴) → (𝑧 ⊆ (𝑔𝑤) → 𝑧 ⊆ (𝐺𝑤)))
226225reximia 3080 . . . . . . 7 (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤) → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
227226ralimi 3082 . . . . . 6 (∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
228227ad2antlr 724 . . . . 5 (((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔𝑤)) ∧ 𝐴 ∈ On) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤))
229 fnex 7221 . . . . . . 7 ((𝐺 Fn (cf‘𝐴) ∧ (cf‘𝐴) ∈ On) → 𝐺 ∈ V)
230185, 2, 229mp2an 689 . . . . . 6 𝐺 ∈ V
231 feq1 6698 . . . . . . 7 (𝑓 = 𝐺 → (𝑓:(cf‘𝐴)⟶𝐴𝐺:(cf‘𝐴)⟶𝐴))
232 smoeq 8354 . . . . . . 7 (𝑓 = 𝐺 → (Smo 𝑓 ↔ Smo 𝐺))
233 fveq1 6890 . . . . . . . . . 10 (𝑓 = 𝐺 → (𝑓𝑤) = (𝐺𝑤))
234233sseq2d 4014 . . . . . . . . 9 (𝑓 = 𝐺 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝐺𝑤)))
235234rexbidv 3177 . . . . . . . 8 (𝑓 = 𝐺 → (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤) ↔ ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)))
236235ralbidv 3176 . . . . . . 7 (𝑓 = 𝐺 → (∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤) ↔ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺𝑤)))
237231, 232, 2363anbi123d 1435 . . . . . 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 3596 . . . . 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 1370 . . . 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 413 . . 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 1935 . 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 395  wo 844  w3a 1086   = wceq 1540  wex 1780  wcel 2105  wral 3060  wrex 3069  Vcvv 3473  cun 3946  wss 3948   ciun 4997  cmpt 5231  dom cdm 5676  cres 5678  Ord word 6363  Oncon0 6364  suc csuc 6366  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1wf1 6540  cfv 6543  Smo wsmo 8349  recscrecs 8374  cfccf 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-smo 8350  df-recs 8375  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-card 9938  df-cf 9940  df-acn 9941
This theorem is referenced by:  cfsmo  10270
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