Theorem wfr1OLD 8327

Description: Obsolete proof of wfr1 8335 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 8319	. 2 ⊢ Fun 𝐹
5		eqid 2733	. . 3 ⊢ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
6	1, 2, 3, 5	wfrlem16OLD 8324	. 2 ⊢ dom 𝐹 = 𝐴
7		df-fn 6547	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
8	4, 6, 7	mpbir2an 710	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1542   ∪ cun 3947  {csn 4629  ⟨cop 4635   Se wse 5630   We wwe 5631  dom cdm 5677   ↾ cres 5679  Predcpred 6300  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544  wrecscwrecs 8296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297

This theorem is referenced by: (None)