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Theorem tfrexlem 5978
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 5205 . . . . 5 (𝑧 = 𝐶 → (recs(𝐹)‘𝑧) = (recs(𝐹)‘𝐶))
21eleq1d 2122 . . . 4 (𝑧 = 𝐶 → ((recs(𝐹)‘𝑧) ∈ V ↔ (recs(𝐹)‘𝐶) ∈ V))
32imbi2d 223 . . 3 (𝑧 = 𝐶 → ((𝜑 → (recs(𝐹)‘𝑧) ∈ V) ↔ (𝜑 → (recs(𝐹)‘𝐶) ∈ V)))
4 inss2 3185 . . . . . . 7 (suc suc 𝑧 ∩ On) ⊆ On
5 ssorduni 4240 . . . . . . 7 ((suc suc 𝑧 ∩ On) ⊆ On → Ord (suc suc 𝑧 ∩ On))
64, 5ax-mp 7 . . . . . 6 Ord (suc suc 𝑧 ∩ On)
7 vex 2577 . . . . . . . . . 10 𝑧 ∈ V
87sucex 4252 . . . . . . . . 9 suc 𝑧 ∈ V
98sucex 4252 . . . . . . . 8 suc suc 𝑧 ∈ V
109inex1 3918 . . . . . . 7 (suc suc 𝑧 ∩ On) ∈ V
1110uniex 4201 . . . . . 6 (suc suc 𝑧 ∩ On) ∈ V
12 elon2 4140 . . . . . 6 ( (suc suc 𝑧 ∩ On) ∈ On ↔ (Ord (suc suc 𝑧 ∩ On) ∧ (suc suc 𝑧 ∩ On) ∈ V))
136, 11, 12mpbir2an 860 . . . . 5 (suc suc 𝑧 ∩ On) ∈ On
14 tfrexlem.1 . . . . . . 7 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1514tfrlem3 5956 . . . . . 6 𝐴 = {𝑣 ∣ ∃𝑧 ∈ On (𝑣 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑣𝑢) = (𝐹‘(𝑣𝑢)))}
16 tfrexlem.2 . . . . . . 7 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
17 fveq2 5205 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1817eleq1d 2122 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑧) ∈ V))
1918anbi2d 445 . . . . . . . 8 (𝑥 = 𝑧 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑧) ∈ V)))
2019cbvalv 1810 . . . . . . 7 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2116, 20sylib 131 . . . . . 6 (𝜑 → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2215, 21tfrlemi1 5976 . . . . 5 ((𝜑 (suc suc 𝑧 ∩ On) ∈ On) → ∃𝑔(𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2313, 22mpan2 409 . . . 4 (𝜑 → ∃𝑔(𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2415recsfval 5961 . . . . . . . . . . 11 recs(𝐹) = 𝐴
2524breqi 3797 . . . . . . . . . 10 (𝑧recs(𝐹)𝑦𝑧 𝐴𝑦)
26 df-br 3792 . . . . . . . . . 10 (𝑧 𝐴𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴)
27 eluni 3610 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
2825, 26, 273bitri 199 . . . . . . . . 9 (𝑧recs(𝐹)𝑦 ↔ ∃(⟨𝑧, 𝑦⟩ ∈ 𝐴))
297sucid 4181 . . . . . . . . . . . . . . . . 17 𝑧 ∈ suc 𝑧
30 simpr 107 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝐴)
31 vex 2577 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∈ V
3214, 31tfrlem3a 5955 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ↔ ∃𝑡 ∈ On ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))
3330, 32sylib 131 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → ∃𝑡 ∈ On ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))
34 simprl 491 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑡 ∈ On)
35 simprrl 499 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → Fn 𝑡)
36 simpll 489 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → ⟨𝑧, 𝑦⟩ ∈ )
37 fnop 5029 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (( Fn 𝑡 ∧ ⟨𝑧, 𝑦⟩ ∈ ) → 𝑧𝑡)
3835, 36, 37syl2anc 397 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑧𝑡)
39 onelon 4148 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ On ∧ 𝑧𝑡) → 𝑧 ∈ On)
4034, 38, 39syl2anc 397 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((⟨𝑧, 𝑦⟩ ∈ 𝐴) ∧ (𝑡 ∈ On ∧ ( Fn 𝑡 ∧ ∀𝑒𝑡 (𝑒) = (𝐹‘(𝑒))))) → 𝑧 ∈ On)
4133, 40rexlimddv 2454 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑧 ∈ On)
4241adantl 266 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 ∈ On)
43 suceloni 4254 . . . . . . . . . . . . . . . . . . . . . . 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 4254 . . . . . . . . . . . . . . . . . . . . . 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 4246 . . . . . . . . . . . . . . . . . . . . 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 2958 . . . . . . . . . . . . . . . . . . . 20 (suc suc 𝑧 ⊆ On ↔ (suc suc 𝑧 ∩ On) = suc suc 𝑧)
5048, 49sylib 131 . . . . . . . . . . . . . . . . . . 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 3618 . . . . . . . . . . . . . . . . . 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 4139 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑧 ∈ On → Ord suc 𝑧)
53 ordtr 4142 . . . . . . . . . . . . . . . . . . . 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 4177 . . . . . . . . . . . . . . . . . . 19 (Tr suc 𝑧 suc suc 𝑧 = suc 𝑧)
5654, 55sylib 131 . . . . . . . . . . . . . . . . . 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 2088 . . . . . . . . . . . . . . . . 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 2143 . . . . . . . . . . . . . . . 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 5025 . . . . . . . . . . . . . . . . 17 (𝑔 Fn (suc suc 𝑧 ∩ On) → dom 𝑔 = (suc suc 𝑧 ∩ On))
6059ad2antrr 465 . . . . . . . . . . . . . . . 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 2133 . . . . . . . . . . . . . . 15 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧 ∈ dom 𝑔)
627eldm 4559 . . . . . . . . . . . . . . 15 (𝑧 ∈ dom 𝑔 ↔ ∃𝑥 𝑧𝑔𝑥)
6361, 62sylib 131 . . . . . . . . . . . . . 14 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → ∃𝑥 𝑧𝑔𝑥)
64 simpr 107 . . . . . . . . . . . . . . 15 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑔𝑥)
65 fneq2 5015 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (suc suc 𝑧 ∩ On) → (𝑔 Fn 𝑣𝑔 Fn (suc suc 𝑧 ∩ On)))
66 raleq 2522 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (suc suc 𝑧 ∩ On) → (∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤)) ↔ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))))
6765, 66anbi12d 450 . . . . . . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . 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 2577 . . . . . . . . . . . . . . . . . . 19 𝑔 ∈ V
7114, 70tfrlem3a 5955 . . . . . . . . . . . . . . . . . 18 (𝑔𝐴 ↔ ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑤𝑣 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
7269, 71sylibr 141 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑔𝐴)
7372ad2antrr 465 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑔𝐴)
74 simplrr 496 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝐴)
75 simplrl 495 . . . . . . . . . . . . . . . . 17 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → ⟨𝑧, 𝑦⟩ ∈ )
76 df-br 3792 . . . . . . . . . . . . . . . . 17 (𝑧𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ )
7775, 76sylibr 141 . . . . . . . . . . . . . . . 16 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑦)
7815tfrlem5 5960 . . . . . . . . . . . . . . . . 17 ((𝑔𝐴𝐴) → ((𝑧𝑔𝑥𝑧𝑦) → 𝑥 = 𝑦))
7978imp 119 . . . . . . . . . . . . . . . 16 (((𝑔𝐴𝐴) ∧ (𝑧𝑔𝑥𝑧𝑦)) → 𝑥 = 𝑦)
8073, 74, 64, 77, 79syl22anc 1147 . . . . . . . . . . . . . . 15 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑥 = 𝑦)
8164, 80breqtrd 3815 . . . . . . . . . . . . . 14 ((((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) ∧ 𝑧𝑔𝑥) → 𝑧𝑔𝑦)
8263, 81exlimddv 1794 . . . . . . . . . . . . 13 (((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) ∧ (⟨𝑧, 𝑦⟩ ∈ 𝐴)) → 𝑧𝑔𝑦)
83 vex 2577 . . . . . . . . . . . . . 14 𝑦 ∈ V
847, 83brelrn 4594 . . . . . . . . . . . . 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 3635 . . . . . . . . . . . 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 112 . . . . . . . . . 10 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ((⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑦 ran 𝑔))
8988exlimdv 1716 . . . . . . . . 9 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (∃(⟨𝑧, 𝑦⟩ ∈ 𝐴) → 𝑦 ran 𝑔))
9028, 89syl5bi 145 . . . . . . . 8 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (𝑧recs(𝐹)𝑦𝑦 ran 𝑔))
9190alrimiv 1770 . . . . . . 7 ((𝑔 Fn (suc suc 𝑧 ∩ On) ∧ ∀𝑤 (suc suc 𝑧 ∩ On)(𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑦(𝑧recs(𝐹)𝑦𝑦 ran 𝑔))
92 fvss 5216 . . . . . . 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 4626 . . . . . . . 8 ran 𝑔 ∈ V
9594uniex 4201 . . . . . . 7 ran 𝑔 ∈ V
9695ssex 3921 . . . . . 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 1505 . . . 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 2630 . 2 (𝐶𝑉 → (𝜑 → (recs(𝐹)‘𝐶) ∈ V))
101100impcom 120 1 ((𝜑𝐶𝑉) → (recs(𝐹)‘𝐶) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wral 2323  wrex 2324  Vcvv 2574  cin 2943  wss 2944  cop 3405   cuni 3607   class class class wbr 3791  Tr wtr 3881  Ord word 4126  Oncon0 4127  suc csuc 4129  dom cdm 4372  ran crn 4373  cres 4374  Fun wfun 4923   Fn wfn 4924  cfv 4929  recscrecs 5949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-recs 5950
This theorem is referenced by:  tfrex  5984
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