MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfr1OLD Structured version   Visualization version   GIF version

Theorem wfr1OLD 8273
Description: Obsolete proof of wfr1 8281 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.
StepHypRef Expression
1 wfr1OLD.1 . . 3 𝑅 We 𝐴
2 wfr1OLD.2 . . 3 𝑅 Se 𝐴
3 wfr1OLD.3 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
41, 2, 3wfrfunOLD 8265 . 2 Fun 𝐹
5 eqid 2736 . . 3 (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
61, 2, 3, 5wfrlem16OLD 8270 . 2 dom 𝐹 = 𝐴
7 df-fn 6499 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
84, 6, 7mpbir2an 709 1 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cun 3908  {csn 4586  cop 4592   Se wse 5586   We wwe 5587  dom cdm 5633  cres 5635  Predcpred 6252  Fun wfun 6490   Fn wfn 6491  cfv 6496  wrecscwrecs 8242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator