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Theorem f1dom2g 8964
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8967 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) (Proof shortened by BTernaryTau, 25-Sep-2024.)
Assertion
Ref Expression
f1dom2g ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Proof of Theorem f1dom2g
StepHypRef Expression
1 f1f 6780 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7920 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1160 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1124 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹 ∈ V)
5 f1dom3g 8962 . 2 ((𝐹 ∈ V ∧ 𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
64, 5syld3an1 1407 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  Vcvv 3468   class class class wbr 5141  wf 6532  1-1wf1 6533  cdom 8936
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  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-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-dom 8940
This theorem is referenced by:  ssdomg  8995  domdifsn  9053  sucdom2OLD  9081  unxpdomlem3  9251  unbnn  9298  fodomacn  10050  hauspwpwdom  23843
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