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Theorem tfr2a 8435
Description: A weak version of tfr2 8438 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 2737 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem9 8425 . . 3 (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43dmeqi 5915 . . 3 dom 𝐹 = dom recs(𝐺)
52, 4eleq2s 2859 . 2 (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
63fveq1i 6907 . 2 (𝐹𝐴) = (recs(𝐺)‘𝐴)
73reseq1i 5993 . . 3 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87fveq2i 6909 . 2 (𝐺‘(𝐹𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴))
95, 6, 83eqtr4g 2802 1 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  dom cdm 5685  cres 5687  Oncon0 6384   Fn wfn 6556  cfv 6561  recscrecs 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411
This theorem is referenced by:  tfr2  8438  rdgvalg  8459  ordtypelem3  9560
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