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Theorem tfrexlem 6353
Description: The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
tfrexlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrexlem.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
Assertion
Ref Expression
tfrexlem ((𝜑𝐶𝑉) → (recs(𝐹)‘𝐶) ∈ V)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐶(𝑥,𝑦,𝑓)   𝑉(𝑥,𝑦,𝑓)

Proof of Theorem tfrexlem
Dummy variables 𝑒 𝑔 𝑢 𝑣 𝑡 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5530 . . . . 5 (𝑧 = 𝐶 → (recs(𝐹)‘𝑧) = (recs(𝐹)‘𝐶))
21eleq1d 2258 . . . 4 (𝑧 = 𝐶 → ((recs(𝐹)‘𝑧) ∈ V ↔ (recs(𝐹)‘𝐶) ∈ V))
32imbi2d 230 . . 3 (𝑧 = 𝐶 → ((𝜑 → (recs(𝐹)‘𝑧) ∈ V) ↔ (𝜑 → (recs(𝐹)‘𝐶) ∈ V)))
4 inss2 3371 . . . . . . 7 (suc suc 𝑧 ∩ On) ⊆ On
5 ssorduni 4501 . . . . . . 7 ((suc suc 𝑧 ∩ On) ⊆ On → Ord (suc suc 𝑧 ∩ On))
64, 5ax-mp 5 . . . . . 6 Ord (suc suc 𝑧 ∩ On)
7 vex 2755 . . . . . . . . . 10 𝑧 ∈ V
87sucex 4513 . . . . . . . . 9 suc 𝑧 ∈ V
98sucex 4513 . . . . . . . 8 suc suc 𝑧 ∈ V
109inex1 4152 . . . . . . 7 (suc suc 𝑧 ∩ On) ∈ V
1110uniex 4452 . . . . . 6 (suc suc 𝑧 ∩ On) ∈ V
12 elon2 4391 . . . . . 6 ( (suc suc 𝑧 ∩ On) ∈ On ↔ (Ord (suc suc 𝑧 ∩ On) ∧ (suc suc 𝑧 ∩ On) ∈ V))
136, 11, 12mpbir2an 944 . . . . 5 (suc suc 𝑧 ∩ On) ∈ On
14 tfrexlem.1 . . . . . . 7 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1514tfrlem3 6330 . . . . . 6 𝐴 = {𝑣 ∣ ∃𝑧 ∈ On (𝑣 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑣𝑢) = (𝐹‘(𝑣𝑢)))}
16 tfrexlem.2 . . . . . . 7 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
17 fveq2 5530 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1817eleq1d 2258 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑧) ∈ V))
1918anbi2d 464 . . . . . . . 8 (𝑥 = 𝑧 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑧) ∈ V)))
2019cbvalv 1929 . . . . . . 7 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2116, 20sylib 122 . . . . . 6 (𝜑 → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2215, 21tfrlemi1 6351 . . . . 5 ((𝜑 (suc suc 𝑧 ∩ On) ∈ On) → ∃𝑔(𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2313, 22mpan2 425 . . . 4 (𝜑 → ∃𝑔(𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2415recsfval 6334 . . . . . . . . . . 11 recs(𝐹) = 𝐴
2524breqi 4024 . . . . . . . . . 10 (𝑧recs(𝐹)𝑦𝑧 𝐴𝑦)
26 df-br 4019 . . . . . . . . . 10 (𝑧 𝐴𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴)
27 eluni 3827 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
2825, 26, 273bitri 206 . . . . . . . . 9 (𝑧recs(𝐹)𝑦 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
297sucid 4432 . . . . . . . . . . . . . . . . 17 𝑧 ∈ suc 𝑧
30 simpr 110 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝐴)
31 vex 2755 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∈ V
3214, 31tfrlem3a 6329 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ↔ ∃𝑡 ∈ On ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))
3330, 32sylib 122 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → ∃𝑡 ∈ On ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))
34 simprl 529 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑡 ∈ On)
35 simprrl 539 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → Fn 𝑡)
36 simpll 527 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → ⟨𝑧, 𝑦⟩ ∈ )
37 fnop 5334 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (( Fn 𝑡 ∧ ⟨𝑧, 𝑦⟩ ∈ ) → 𝑧𝑡)
3835, 36, 37syl2anc 411 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑧𝑡)
39 onelon 4399 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ On ∧ 𝑧𝑡) → 𝑧 ∈ On)
4034, 38, 39syl2anc 411 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑧 ∈ On)
4133, 40rexlimddv 2612 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑧 ∈ On)
4241adantl 277 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 ∈ On)
43 onsuc 4515 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ On → suc 𝑧 ∈ On)
4442, 43syl 14 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → suc 𝑧 ∈ On)
45 onsuc 4515 . . . . . . . . . . . . . . . . . . . . . 22 (suc 𝑧 ∈ On → suc suc 𝑧 ∈ On)
4644, 45syl 14 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → suc suc 𝑧 ∈ On)
47 onss 4507 . . . . . . . . . . . . . . . . . . . . 21 (suc suc 𝑧 ∈ On → suc suc 𝑧 ⊆ On)
4846, 47syl 14 . . . . . . . . . . . . . . . . . . . 20 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → suc suc 𝑧 ⊆ On)
49 df-ss 3157 . . . . . . . . . . . . . . . . . . . 20 (suc suc 𝑧 ⊆ On ↔ (suc suc 𝑧 ∩ On) = suc suc 𝑧)
5048, 49sylib 122 . . . . . . . . . . . . . . . . . . 19 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → (suc suc 𝑧 ∩ On) = suc suc 𝑧)
5150unieqd 3835 . . . . . . . . . . . . . . . . . 18 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → (suc suc 𝑧 ∩ On) = suc suc 𝑧)
52 eloni 4390 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑧 ∈ On → Ord suc 𝑧)
53 ordtr 4393 . . . . . . . . . . . . . . . . . . . 20 (Ord suc 𝑧 → Tr suc 𝑧)
5444, 52, 533syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → Tr suc 𝑧)
558unisuc 4428 . . . . . . . . . . . . . . . . . . 19 (Tr suc 𝑧 suc suc 𝑧 = suc 𝑧)
5654, 55sylib 122 . . . . . . . . . . . . . . . . . 18 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → suc suc 𝑧 = suc 𝑧)
5751, 56eqtrd 2222 . . . . . . . . . . . . . . . . 17 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → (suc suc 𝑧 ∩ On) = suc 𝑧)
5829, 57eleqtrrid 2279 . . . . . . . . . . . . . . . 16 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 (suc suc 𝑧 ∩ On))
59 fndm 5330 . . . . . . . . . . . . . . . . 17 (𝑔 Fn (suc suc 𝑧 ∩ On) → dom 𝑔 = (suc suc 𝑧 ∩ On))
6059ad2antrr 488 . . . . . . . . . . . . . . . 16 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → dom 𝑔 = (suc suc 𝑧 ∩ On))
6158, 60eleqtrrd 2269 . . . . . . . . . . . . . . 15 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 ∈ dom 𝑔)
627eldm 4839 . . . . . . . . . . . . . . 15 (𝑧 ∈ dom 𝑔 ↔ ∃𝑥 𝑧𝑔𝑥)
6361, 62sylib 122 . . . . . . . . . . . . . 14 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → ∃𝑥 𝑧𝑔𝑥)
64 simpr 110 . . . . . . . . . . . . . . 15 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑔𝑥)
65 fneq2 5320 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (suc suc 𝑧 ∩ On) → (𝑔 Fn 𝑣𝑔 Fn (suc suc 𝑧 ∩ On)))
66 raleq 2686 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (suc suc 𝑧 ∩ On) → (∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤)) ↔ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
6765, 66anbi12d 473 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (suc suc 𝑧 ∩ On) → ((𝑔 Fn 𝑣 ∧ ∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) ↔ (𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤)))))
6867rspcev 2856 . . . . . . . . . . . . . . . . . . 19 (( (suc suc 𝑧 ∩ On) ∈ On ∧ (𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
6913, 68mpan 424 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
70 vex 2755 . . . . . . . . . . . . . . . . . . 19 𝑔 ∈ V
7114, 70tfrlem3a 6329 . . . . . . . . . . . . . . . . . 18 (𝑔𝐴 ↔ ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
7269, 71sylibr 134 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑔𝐴)
7372ad2antrr 488 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑔𝐴)
74 simplrr 536 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝐴)
75 simplrl 535 . . . . . . . . . . . . . . . . 17 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → ⟨𝑧, 𝑦⟩ ∈ )
76 df-br 4019 . . . . . . . . . . . . . . . . 17 (𝑧𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ )
7775, 76sylibr 134 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑦)
7815tfrlem5 6333 . . . . . . . . . . . . . . . . 17 ((𝑔𝐴𝐴) → ((𝑧𝑔𝑥𝑧𝑦) → 𝑥 = 𝑦))
7978imp 124 . . . . . . . . . . . . . . . 16 (((𝑔𝐴𝐴) ∧ (𝑧𝑔𝑥𝑧𝑦)) → 𝑥 = 𝑦)
8073, 74, 64, 77, 79syl22anc 1250 . . . . . . . . . . . . . . 15 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑥 = 𝑦)
8164, 80breqtrd 4044 . . . . . . . . . . . . . 14 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑔𝑦)
8263, 81exlimddv 1910 . . . . . . . . . . . . 13 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧𝑔𝑦)
83 vex 2755 . . . . . . . . . . . . . 14 𝑦 ∈ V
847, 83brelrn 4875 . . . . . . . . . . . . 13 (𝑧𝑔𝑦𝑦 ∈ ran 𝑔)
8582, 84syl 14 . . . . . . . . . . . 12 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑦 ∈ ran 𝑔)
86 elssuni 3852 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝑔𝑦 ran 𝑔)
8785, 86syl 14 . . . . . . . . . . 11 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑦 ran 𝑔)
8887ex 115 . . . . . . . . . 10 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑦 ran 𝑔))
8988exlimdv 1830 . . . . . . . . 9 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (∃(⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑦 ran 𝑔))
9028, 89biimtrid 152 . . . . . . . 8 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (𝑧recs(𝐹)𝑦𝑦 ran 𝑔))
9190alrimiv 1885 . . . . . . 7 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑦(𝑧recs(𝐹)𝑦𝑦 ran 𝑔))
92 fvss 5544 . . . . . . 7 (∀𝑦(𝑧recs(𝐹)𝑦𝑦 ran 𝑔) → (recs(𝐹)‘𝑧) ⊆ ran 𝑔)
9391, 92syl 14 . . . . . 6 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (recs(𝐹)‘𝑧) ⊆ ran 𝑔)
9470rnex 4909 . . . . . . . 8 ran 𝑔 ∈ V
9594uniex 4452 . . . . . . 7 ran 𝑔 ∈ V
9695ssex 4155 . . . . . 6 ((recs(𝐹)‘𝑧) ⊆ ran 𝑔 → (recs(𝐹)‘𝑧) ∈ V)
9793, 96syl 14 . . . . 5 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (recs(𝐹)‘𝑧) ∈ V)
9897exlimiv 1609 . . . 4 (∃𝑔(𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (recs(𝐹)‘𝑧) ∈ V)
9923, 98syl 14 . . 3 (𝜑 → (recs(𝐹)‘𝑧) ∈ V)
1003, 99vtoclg 2812 . 2 (𝐶𝑉 → (𝜑 → (recs(𝐹)‘𝐶) ∈ V))
101100impcom 125 1 ((𝜑𝐶𝑉) → (recs(𝐹)‘𝐶) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1362   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wral 2468  wrex 2469  Vcvv 2752  cin 3143  wss 3144  cop 3610   cuni 3824   class class class wbr 4018  Tr wtr 4116  Ord word 4377  Oncon0 4378  suc csuc 4380  dom cdm 4641  ran crn 4642  cres 4643  Fun wfun 5225   Fn wfn 5226  cfv 5231  recscrecs 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-recs 6324
This theorem is referenced by:  tfrex  6387
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