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Theorem f1dom3g 9007
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 9011 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 6800 . . . . 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 1130 . 2 ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
5 brdomg 8996 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
653ad2ant2 1133 . 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 1086  wex 1776  wcel 2106   class class class wbr 5148  1-1wf1 6560  cdom 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-dom 8986
This theorem is referenced by:  f1dom2g  9009  undom  9098  f1domfi  9219  f1domfi2  9220  sucdom2  9241
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