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Theorem tfrex 6149
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 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
Assertion
Ref Expression
tfrex ((𝜑𝐴𝑉) → (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 5321 . 2 (𝐹𝐴) = (recs(𝐺)‘𝐴)
3 eqid 2089 . . . 4 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))}
43tfrlem3 6092 . . 3 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
5 tfrex.2 . . 3 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
64, 5tfrexlem 6115 . 2 ((𝜑𝐴𝑉) → (recs(𝐺)‘𝐴) ∈ V)
72, 6syl5eqel 2175 1 ((𝜑𝐴𝑉) → (𝐹𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1288   = wceq 1290  wcel 1439  {cab 2075  wral 2360  wrex 2361  Vcvv 2622  Oncon0 4201  cres 4456  Fun wfun 5024   Fn wfn 5025  cfv 5030  recscrecs 6085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-iord 4204  df-on 4206  df-suc 4209  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-recs 6086
This theorem is referenced by:  rdgexggg  6158
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