Theorem wfr1OLD 8325

Description: Obsolete version of wfr1 8333 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 8317	. 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 8322	. 2 ⊢ dom 𝐹 = 𝐴
7		df-fn 6539	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
8	4, 6, 7	mpbir2an 708	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1533   ∪ cun 3941  {csn 4623  ⟨cop 4629   Se wse 5622   We wwe 5623  dom cdm 5669   ↾ cres 5671  Predcpred 6292  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  wrecscwrecs 8294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-2nd 7972  df-frecs 8264  df-wrecs 8295

This theorem is referenced by: (None)