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Theorem f1dom2g 8948
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8951 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 6774 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7906 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1163 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1127 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹 ∈ V)
5 f1dom3g 8946 . 2 ((𝐹 ∈ V ∧ 𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
64, 5syld3an1 1410 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  Vcvv 3473   class class class wbr 5141  wf 6528  1-1wf1 6529  cdom 8920
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-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6534  df-fn 6535  df-f 6536  df-f1 6537  df-dom 8924
This theorem is referenced by:  ssdomg  8979  domdifsn  9037  sucdom2OLD  9065  unxpdomlem3  9235  unbnn  9282  fodomacn  10033  hauspwpwdom  23421
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