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Theorem tfrexlem 6031
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 5253 . . . . 5 (𝑧 = 𝐶 → (recs(𝐹)‘𝑧) = (recs(𝐹)‘𝐶))
21eleq1d 2151 . . . 4 (𝑧 = 𝐶 → ((recs(𝐹)‘𝑧) ∈ V ↔ (recs(𝐹)‘𝐶) ∈ V))
32imbi2d 228 . . 3 (𝑧 = 𝐶 → ((𝜑 → (recs(𝐹)‘𝑧) ∈ V) ↔ (𝜑 → (recs(𝐹)‘𝐶) ∈ V)))
4 inss2 3205 . . . . . . 7 (suc suc 𝑧 ∩ On) ⊆ On
5 ssorduni 4267 . . . . . . 7 ((suc suc 𝑧 ∩ On) ⊆ On → Ord (suc suc 𝑧 ∩ On))
64, 5ax-mp 7 . . . . . 6 Ord (suc suc 𝑧 ∩ On)
7 vex 2615 . . . . . . . . . 10 𝑧 ∈ V
87sucex 4279 . . . . . . . . 9 suc 𝑧 ∈ V
98sucex 4279 . . . . . . . 8 suc suc 𝑧 ∈ V
109inex1 3938 . . . . . . 7 (suc suc 𝑧 ∩ On) ∈ V
1110uniex 4228 . . . . . 6 (suc suc 𝑧 ∩ On) ∈ V
12 elon2 4167 . . . . . 6 ( (suc suc 𝑧 ∩ On) ∈ On ↔ (Ord (suc suc 𝑧 ∩ On) ∧ (suc suc 𝑧 ∩ On) ∈ V))
136, 11, 12mpbir2an 884 . . . . 5 (suc suc 𝑧 ∩ On) ∈ On
14 tfrexlem.1 . . . . . . 7 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1514tfrlem3 6008 . . . . . 6 𝐴 = {𝑣 ∣ ∃𝑧 ∈ On (𝑣 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑣𝑢) = (𝐹‘(𝑣𝑢)))}
16 tfrexlem.2 . . . . . . 7 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
17 fveq2 5253 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1817eleq1d 2151 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑧) ∈ V))
1918anbi2d 452 . . . . . . . 8 (𝑥 = 𝑧 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑧) ∈ V)))
2019cbvalv 1837 . . . . . . 7 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2116, 20sylib 120 . . . . . 6 (𝜑 → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2215, 21tfrlemi1 6029 . . . . 5 ((𝜑 (suc suc 𝑧 ∩ On) ∈ On) → ∃𝑔(𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2313, 22mpan2 416 . . . 4 (𝜑 → ∃𝑔(𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2415recsfval 6012 . . . . . . . . . . 11 recs(𝐹) = 𝐴
2524breqi 3817 . . . . . . . . . 10 (𝑧recs(𝐹)𝑦𝑧 𝐴𝑦)
26 df-br 3812 . . . . . . . . . 10 (𝑧 𝐴𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴)
27 eluni 3630 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
2825, 26, 273bitri 204 . . . . . . . . 9 (𝑧recs(𝐹)𝑦 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
297sucid 4208 . . . . . . . . . . . . . . . . 17 𝑧 ∈ suc 𝑧
30 simpr 108 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝐴)
31 vex 2615 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∈ V
3214, 31tfrlem3a 6007 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ↔ ∃𝑡 ∈ On ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))
3330, 32sylib 120 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → ∃𝑡 ∈ On ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))
34 simprl 498 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑡 ∈ On)
35 simprrl 506 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → Fn 𝑡)
36 simpll 496 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → ⟨𝑧, 𝑦⟩ ∈ )
37 fnop 5070 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (( Fn 𝑡 ∧ ⟨𝑧, 𝑦⟩ ∈ ) → 𝑧𝑡)
3835, 36, 37syl2anc 403 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑧𝑡)
39 onelon 4175 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ On ∧ 𝑧𝑡) → 𝑧 ∈ On)
4034, 38, 39syl2anc 403 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑧 ∈ On)
4133, 40rexlimddv 2487 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑧 ∈ On)
4241adantl 271 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 ∈ On)
43 suceloni 4281 . . . . . . . . . . . . . . . . . . . . . . 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 suceloni 4281 . . . . . . . . . . . . . . . . . . . . . 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 4273 . . . . . . . . . . . . . . . . . . . . 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 2997 . . . . . . . . . . . . . . . . . . . 20 (suc suc 𝑧 ⊆ On ↔ (suc suc 𝑧 ∩ On) = suc suc 𝑧)
5048, 49sylib 120 . . . . . . . . . . . . . . . . . . 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 3638 . . . . . . . . . . . . . . . . . 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 4166 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑧 ∈ On → Ord suc 𝑧)
53 ordtr 4169 . . . . . . . . . . . . . . . . . . . 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 4204 . . . . . . . . . . . . . . . . . . 19 (Tr suc 𝑧 suc suc 𝑧 = suc 𝑧)
5654, 55sylib 120 . . . . . . . . . . . . . . . . . 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 2115 . . . . . . . . . . . . . . . . 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, 57syl5eleqr 2172 . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . 17 (𝑔 Fn (suc suc 𝑧 ∩ On) → dom 𝑔 = (suc suc 𝑧 ∩ On))
6059ad2antrr 472 . . . . . . . . . . . . . . . 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 2162 . . . . . . . . . . . . . . 15 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 ∈ dom 𝑔)
627eldm 4591 . . . . . . . . . . . . . . 15 (𝑧 ∈ dom 𝑔 ↔ ∃𝑥 𝑧𝑔𝑥)
6361, 62sylib 120 . . . . . . . . . . . . . 14 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → ∃𝑥 𝑧𝑔𝑥)
64 simpr 108 . . . . . . . . . . . . . . 15 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑔𝑥)
65 fneq2 5056 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (suc suc 𝑧 ∩ On) → (𝑔 Fn 𝑣𝑔 Fn (suc suc 𝑧 ∩ On)))
66 raleq 2555 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (suc suc 𝑧 ∩ On) → (∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤)) ↔ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
6765, 66anbi12d 457 . . . . . . . . . . . . . . . . . . . 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 2712 . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . 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 2615 . . . . . . . . . . . . . . . . . . 19 𝑔 ∈ V
7114, 70tfrlem3a 6007 . . . . . . . . . . . . . . . . . 18 (𝑔𝐴 ↔ ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
7269, 71sylibr 132 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑔𝐴)
7372ad2antrr 472 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑔𝐴)
74 simplrr 503 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝐴)
75 simplrl 502 . . . . . . . . . . . . . . . . 17 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → ⟨𝑧, 𝑦⟩ ∈ )
76 df-br 3812 . . . . . . . . . . . . . . . . 17 (𝑧𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ )
7775, 76sylibr 132 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑦)
7815tfrlem5 6011 . . . . . . . . . . . . . . . . 17 ((𝑔𝐴𝐴) → ((𝑧𝑔𝑥𝑧𝑦) → 𝑥 = 𝑦))
7978imp 122 . . . . . . . . . . . . . . . 16 (((𝑔𝐴𝐴) ∧ (𝑧𝑔𝑥𝑧𝑦)) → 𝑥 = 𝑦)
8073, 74, 64, 77, 79syl22anc 1171 . . . . . . . . . . . . . . 15 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑥 = 𝑦)
8164, 80breqtrd 3835 . . . . . . . . . . . . . 14 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑔𝑦)
8263, 81exlimddv 1821 . . . . . . . . . . . . 13 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧𝑔𝑦)
83 vex 2615 . . . . . . . . . . . . . 14 𝑦 ∈ V
847, 83brelrn 4626 . . . . . . . . . . . . 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 3655 . . . . . . . . . . . 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 113 . . . . . . . . . 10 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑦 ran 𝑔))
8988exlimdv 1742 . . . . . . . . 9 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (∃(⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑦 ran 𝑔))
9028, 89syl5bi 150 . . . . . . . 8 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (𝑧recs(𝐹)𝑦𝑦 ran 𝑔))
9190alrimiv 1797 . . . . . . 7 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑦(𝑧recs(𝐹)𝑦𝑦 ran 𝑔))
92 fvss 5264 . . . . . . 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 4658 . . . . . . . 8 ran 𝑔 ∈ V
9594uniex 4228 . . . . . . 7 ran 𝑔 ∈ V
9695ssex 3941 . . . . . 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 1530 . . . 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 2669 . 2 (𝐶𝑉 → (𝜑 → (recs(𝐹)‘𝐶) ∈ V))
101100impcom 123 1 ((𝜑𝐶𝑉) → (recs(𝐹)‘𝐶) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2612  cin 2983  wss 2984  cop 3425   cuni 3627   class class class wbr 3811  Tr wtr 3901  Ord word 4153  Oncon0 4154  suc csuc 4156  dom cdm 4401  ran crn 4402  cres 4403  Fun wfun 4963   Fn wfn 4964  cfv 4969  recscrecs 6001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-recs 6002
This theorem is referenced by:  tfrex  6065
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