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Theorem tfr2a 8217
Description: A weak version of tfr2 8220 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr2a (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))

Proof of Theorem tfr2a
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2740 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem9 8207 . . 3 (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43dmeqi 5812 . . 3 dom 𝐹 = dom recs(𝐺)
52, 4eleq2s 2859 . 2 (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
63fveq1i 6772 . 2 (𝐹𝐴) = (recs(𝐺)‘𝐴)
73reseq1i 5886 . . 3 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87fveq2i 6774 . 2 (𝐺‘(𝐹𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴))
95, 6, 83eqtr4g 2805 1 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110  {cab 2717  wral 3066  wrex 3067  dom cdm 5590  cres 5592  Oncon0 6265   Fn wfn 6427  cfv 6432  recscrecs 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fo 6438  df-fv 6440  df-ov 7274  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193
This theorem is referenced by:  tfr2  8220  rdgvalg  8241  ordtypelem3  9257
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