Theorem tfr2a 8388

Description: A weak version of tfr2 8391 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 2762	. . . 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 8378	. . 3 ⊢ (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
3		tfr.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4	3	dmeqi 5892	. . 3 ⊢ dom 𝐹 = dom recs(𝐺)
5	2, 4	eleq2s 2880	. 2 ⊢ (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
6	3	fveq1i 6883	. 2 ⊢ (𝐹‘𝐴) = (recs(𝐺)‘𝐴)
7	3	reseq1i 5972	. . 3 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴)
8	7	fveq2i 6885	. 2 ⊢ (𝐺‘(𝐹 ↾ 𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴))
9	5, 6, 8	3eqtr4g 2822	1 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  ∃wrex 3088  dom cdm 5659   ↾ cres 5661  Oncon0 6361   Fn wfn 6532  ‘cfv 6537  recscrecs 8363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7420  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364

This theorem is used by:  tfr2  8391  rdgvalg  8412  ordtypelem3  9496