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Theorem f1dom3g 9008
Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 9012 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024.)
Assertion
Ref Expression
f1dom3g ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Proof of Theorem f1dom3g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1eq1 6799 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝐴1-1𝐵𝐹:𝐴1-1𝐵))
21spcegv 3597 . . . 4 (𝐹𝑉 → (𝐹:𝐴1-1𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵))
32imp 406 . . 3 ((𝐹𝑉𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
433adant2 1132 . 2 ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
5 brdomg 8997 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
653ad2ant2 1135 . 2 ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
74, 6mpbird 257 1 ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087  wex 1779  wcel 2108   class class class wbr 5143  1-1wf1 6558  cdom 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-dom 8987
This theorem is referenced by:  f1dom2g  9010  undom  9099  f1domfi  9221  f1domfi2  9222  sucdom2  9243
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