Theorem tfr2a 7378

Description: A weak version of tfr2 7381 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 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Proof of Theorem tfr2a

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem9 7368	. . 3 ⊢ (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
3		tfr.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4	3	dmeqi 5247	. . 3 ⊢ dom 𝐹 = dom recs(𝐺)
5	2, 4	eleq2s 2706	. 2 ⊢ (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
6	3	fveq1i 6104	. 2 ⊢ (𝐹‘𝐴) = (recs(𝐺)‘𝐴)
7	3	reseq1i 5313	. . 3 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴)
8	7	fveq2i 6106	. 2 ⊢ (𝐺‘(𝐹 ↾ 𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴))
9	5, 6, 8	3eqtr4g 2669	1 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  dom cdm 5038   ↾ cres 5040  Oncon0 5640   Fn wfn 5799  ‘cfv 5804  recscrecs 7354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-wrecs 7294  df-recs 7355

This theorem is referenced by:  tfr2  7381  rdgvalg  7402  ordtypelem3  8308