Theorem wfr2OLD 8358

Description: Obsolete version of wfr2 8366 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 2726	. . . 4 ⊢ (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩}) = (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩})
5	1, 2, 3, 4	wfrlem16OLD 8354	. . 3 ⊢ dom 𝐹 = 𝐴
6	5	eleq2i 2818	. 2 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴)
7	1, 2, 3	wfr2aOLD 8356	. 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
8	6, 7	sylbir 234	1 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   ∪ cun 3945  {csn 4633  ⟨cop 4639   Se wse 5635   We wwe 5636  dom cdm 5682   ↾ cres 5684  Predcpred 6311  ‘cfv 6554  wrecscwrecs 8326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-2nd 8004  df-frecs 8296  df-wrecs 8327

This theorem is referenced by: (None)