Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfr2a Structured version   Visualization version   GIF version

Theorem tfr2a 7536
 Description: A weak version of tfr2 7539 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 2651 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem9 7526 . . 3 (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43dmeqi 5357 . . 3 dom 𝐹 = dom recs(𝐺)
52, 4eleq2s 2748 . 2 (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
63fveq1i 6230 . 2 (𝐹𝐴) = (recs(𝐺)‘𝐴)
73reseq1i 5424 . . 3 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87fveq2i 6232 . 2 (𝐺‘(𝐹𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴))
95, 6, 83eqtr4g 2710 1 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  dom cdm 5143   ↾ cres 5145  Oncon0 5761   Fn wfn 5921  ‘cfv 5926  recscrecs 7512 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-wrecs 7452  df-recs 7513 This theorem is referenced by:  tfr2  7539  rdgvalg  7560  ordtypelem3  8466
 Copyright terms: Public domain W3C validator