MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1dom2gOLD Structured version   Visualization version   GIF version

Theorem f1dom2gOLD 8989
Description: Obsolete version of f1dom2g 8988 as of 25-Sep-2024. (Contributed by Mario Carneiro, 24-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
f1dom2gOLD ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Proof of Theorem f1dom2gOLD
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1f 6788 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7939 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1160 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1124 . . 3 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹 ∈ V)
5 simp3 1135 . . 3 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹:𝐴1-1𝐵)
6 f1eq1 6783 . . 3 (𝑓 = 𝐹 → (𝑓:𝐴1-1𝐵𝐹:𝐴1-1𝐵))
74, 5, 6spcedv 3577 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
8 brdomg 8975 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
983ad2ant2 1131 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
107, 9mpbird 256 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084  wex 1773  wcel 2098  Vcvv 3463   class class class wbr 5143  wf 6539  1-1wf1 6540  cdom 8960
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-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-dom 8964
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator