Theorem wfr2OLD 8367

Description: Obsolete version of wfr2 8375 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfr2OLD.1	⊢ 𝑅 We 𝐴
wfr2OLD.2	⊢ 𝑅 Se 𝐴
wfr2OLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfr2OLD	⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2OLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfr2OLD.1	. . . 4 ⊢ 𝑅 We 𝐴
2		wfr2OLD.2	. . . 4 ⊢ 𝑅 Se 𝐴
3		wfr2OLD.3	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		eqid 2735	. . . 4 ⊢ (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩}) = (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩})
5	1, 2, 3, 4	wfrlem16OLD 8363	. . 3 ⊢ dom 𝐹 = 𝐴
6	5	eleq2i 2831	. 2 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴)
7	1, 2, 3	wfr2aOLD 8365	. 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
8	6, 7	sylbir 235	1 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ∪ cun 3961  {csn 4631  ⟨cop 4637   Se wse 5639   We wwe 5640  dom cdm 5689   ↾ cres 5691  Predcpred 6322  ‘cfv 6563  wrecscwrecs 8335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336

This theorem is referenced by: (None)