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Theorem f1dom2g 7933
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 7935 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
f1dom2g ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Proof of Theorem f1dom2g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1f 6068 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7083 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1356 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1269 . . 3 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹 ∈ V)
5 simp3 1061 . . 3 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹:𝐴1-1𝐵)
6 f1eq1 6063 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐴1-1𝐵𝐹:𝐴1-1𝐵))
76spcegv 3284 . . 3 (𝐹 ∈ V → (𝐹:𝐴1-1𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵))
84, 5, 7sylc 65 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
9 brdomg 7925 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
1093ad2ant2 1081 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
118, 10mpbird 247 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036  wex 1701  wcel 1987  Vcvv 3190   class class class wbr 4623  wf 5853  1-1wf1 5854  cdom 7913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-dom 7917
This theorem is referenced by:  ssdomg  7961  domdifsn  8003  sucdom2  8116  unxpdomlem3  8126  unbnn  8176  fodomacn  8839  hauspwpwdom  21732
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