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Theorem tfrex 5876
 Description: The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
tfrex.1 𝐹 = recs(𝐺)
tfrex.2 (φx(Fun 𝐺 (𝐺x) V))
Assertion
Ref Expression
tfrex ((φ A 𝑉) → (𝐹A) V)
Distinct variable group:   x,𝐺
Allowed substitution hints:   φ(x)   A(x)   𝐹(x)   𝑉(x)

Proof of Theorem tfrex
Dummy variables f g u y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrex.1 . . 3 𝐹 = recs(𝐺)
21fveq1i 5104 . 2 (𝐹A) = (recs(𝐺)‘A)
3 eqid 2022 . . . 4 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {gz On (g Fn z u z (gu) = (𝐺‘(gu)))}
43tfrlem3 5848 . . 3 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {fx On (f Fn x y x (fy) = (𝐺‘(fy)))}
5 tfrex.2 . . 3 (φx(Fun 𝐺 (𝐺x) V))
64, 5tfrexlem 5870 . 2 ((φ A 𝑉) → (recs(𝐺)‘A) V)
72, 6syl5eqel 2106 1 ((φ A 𝑉) → (𝐹A) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226   = wceq 1228   ∈ wcel 1374  {cab 2008  ∀wral 2284  ∃wrex 2285  Vcvv 2535  Oncon0 4049   ↾ cres 4274  Fun wfun 4823   Fn wfn 4824  ‘cfv 4829  recscrecs 5841 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-recs 5842 This theorem is referenced by:  rdgexggg  5884
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