Theorem wfr1OLD 8359

Description: Obsolete version of wfr1 8367 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfr1OLD.1	⊢ 𝑅 We 𝐴
wfr1OLD.2	⊢ 𝑅 Se 𝐴
wfr1OLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfr1OLD	⊢ 𝐹 Fn 𝐴

Proof of Theorem wfr1OLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfr1OLD.1	. . 3 ⊢ 𝑅 We 𝐴
2		wfr1OLD.2	. . 3 ⊢ 𝑅 Se 𝐴
3		wfr1OLD.3	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4	1, 2, 3	wfrfunOLD 8351	. 2 ⊢ Fun 𝐹
5		eqid 2726	. . 3 ⊢ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
6	1, 2, 3, 5	wfrlem16OLD 8356	. 2 ⊢ dom 𝐹 = 𝐴
7		df-fn 6559	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
8	4, 6, 7	mpbir2an 709	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1534   ∪ cun 3945  {csn 4633  ⟨cop 4639   Se wse 5637   We wwe 5638  dom cdm 5684   ↾ cres 5686  Predcpred 6313  Fun wfun 6550   Fn wfn 6551  ‘cfv 6556  wrecscwrecs 8328

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 5292  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748

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 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-po 5596  df-so 5597  df-fr 5639  df-se 5640  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-ov 7429  df-2nd 8006  df-frecs 8298  df-wrecs 8329

This theorem is referenced by: (None)