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Theorem tfrexlem 6053
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 5268 . . . . 5 (𝑧 = 𝐶 → (recs(𝐹)‘𝑧) = (recs(𝐹)‘𝐶))
21eleq1d 2153 . . . 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 3210 . . . . . . 7 (suc suc 𝑧 ∩ On) ⊆ On
5 ssorduni 4277 . . . . . . 7 ((suc suc 𝑧 ∩ On) ⊆ On → Ord (suc suc 𝑧 ∩ On))
64, 5ax-mp 7 . . . . . 6 Ord (suc suc 𝑧 ∩ On)
7 vex 2618 . . . . . . . . . 10 𝑧 ∈ V
87sucex 4289 . . . . . . . . 9 suc 𝑧 ∈ V
98sucex 4289 . . . . . . . 8 suc suc 𝑧 ∈ V
109inex1 3948 . . . . . . 7 (suc suc 𝑧 ∩ On) ∈ V
1110uniex 4238 . . . . . 6 (suc suc 𝑧 ∩ On) ∈ V
12 elon2 4177 . . . . . 6 ( (suc suc 𝑧 ∩ On) ∈ On ↔ (Ord (suc suc 𝑧 ∩ On) ∧ (suc suc 𝑧 ∩ On) ∈ V))
136, 11, 12mpbir2an 886 . . . . 5 (suc suc 𝑧 ∩ On) ∈ On
14 tfrexlem.1 . . . . . . 7 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1514tfrlem3 6030 . . . . . 6 𝐴 = {𝑣 ∣ ∃𝑧 ∈ On (𝑣 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑣𝑢) = (𝐹‘(𝑣𝑢)))}
16 tfrexlem.2 . . . . . . 7 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
17 fveq2 5268 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1817eleq1d 2153 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑧) ∈ V))
1918anbi2d 452 . . . . . . . 8 (𝑥 = 𝑧 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑧) ∈ V)))
2019cbvalv 1839 . . . . . . 7 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2116, 20sylib 120 . . . . . 6 (𝜑 → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2215, 21tfrlemi1 6051 . . . . 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 6034 . . . . . . . . . . 11 recs(𝐹) = 𝐴
2524breqi 3826 . . . . . . . . . 10 (𝑧recs(𝐹)𝑦𝑧 𝐴𝑦)
26 df-br 3821 . . . . . . . . . 10 (𝑧 𝐴𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴)
27 eluni 3639 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
2825, 26, 273bitri 204 . . . . . . . . 9 (𝑧recs(𝐹)𝑦 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
297sucid 4218 . . . . . . . . . . . . . . . . 17 𝑧 ∈ suc 𝑧
30 simpr 108 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝐴)
31 vex 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∈ V
3214, 31tfrlem3a 6029 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5082 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4185 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . . . . . . . . . . . . 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 4291 . . . . . . . . . . . . . . . . . . . . . . 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 4291 . . . . . . . . . . . . . . . . . . . . . 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 4283 . . . . . . . . . . . . . . . . . . . . 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 3001 . . . . . . . . . . . . . . . . . . . 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 3647 . . . . . . . . . . . . . . . . . 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 4176 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑧 ∈ On → Ord suc 𝑧)
53 ordtr 4179 . . . . . . . . . . . . . . . . . . . 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 4214 . . . . . . . . . . . . . . . . . . 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 2117 . . . . . . . . . . . . . . . . 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 2174 . . . . . . . . . . . . . . . 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 5078 . . . . . . . . . . . . . . . . 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 2164 . . . . . . . . . . . . . . 15 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 ∈ dom 𝑔)
627eldm 4601 . . . . . . . . . . . . . . 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 5068 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (suc suc 𝑧 ∩ On) → (𝑔 Fn 𝑣𝑔 Fn (suc suc 𝑧 ∩ On)))
66 raleq 2558 . . . . . . . . . . . . . . . . . . . . 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 2715 . . . . . . . . . . . . . . . . . . 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 2618 . . . . . . . . . . . . . . . . . . 19 𝑔 ∈ V
7114, 70tfrlem3a 6029 . . . . . . . . . . . . . . . . . 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 3821 . . . . . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . . . . . 17 ((𝑔𝐴𝐴) → ((𝑧𝑔𝑥𝑧𝑦) → 𝑥 = 𝑦))
7978imp 122 . . . . . . . . . . . . . . . 16 (((𝑔𝐴𝐴) ∧ (𝑧𝑔𝑥𝑧𝑦)) → 𝑥 = 𝑦)
8073, 74, 64, 77, 79syl22anc 1173 . . . . . . . . . . . . . . 15 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑥 = 𝑦)
8164, 80breqtrd 3844 . . . . . . . . . . . . . 14 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑔𝑦)
8263, 81exlimddv 1823 . . . . . . . . . . . . 13 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧𝑔𝑦)
83 vex 2618 . . . . . . . . . . . . . 14 𝑦 ∈ V
847, 83brelrn 4636 . . . . . . . . . . . . 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 3664 . . . . . . . . . . . 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 1744 . . . . . . . . 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 1799 . . . . . . 7 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑦(𝑧recs(𝐹)𝑦𝑦 ran 𝑔))
92 fvss 5282 . . . . . . 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 4668 . . . . . . . 8 ran 𝑔 ∈ V
9594uniex 4238 . . . . . . 7 ran 𝑔 ∈ V
9695ssex 3951 . . . . . 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 1532 . . . 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 2672 . 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 1285   = wceq 1287  wex 1424  wcel 1436  {cab 2071  wral 2355  wrex 2356  Vcvv 2615  cin 2987  wss 2988  cop 3434   cuni 3636   class class class wbr 3820  Tr wtr 3911  Ord word 4163  Oncon0 4164  suc csuc 4166  dom cdm 4411  ran crn 4412  cres 4413  Fun wfun 4975   Fn wfn 4976  cfv 4981  recscrecs 6023
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-recs 6024
This theorem is referenced by:  tfrex  6087
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