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Theorem f1dom3g 8943
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 8948 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 6766 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝐴1-1𝐵𝐹:𝐴1-1𝐵))
21spcegv 3581 . . . 4 (𝐹𝑉 → (𝐹:𝐴1-1𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵))
32imp 407 . . 3 ((𝐹𝑉𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
433adant2 1131 . 2 ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
5 brdomg 8932 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
653ad2ant2 1134 . 2 ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
74, 6mpbird 256 1 ((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087  wex 1781  wcel 2106   class class class wbr 5138  1-1wf1 6526  cdom 8917
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 5289  ax-nul 5296  ax-pr 5417  ax-un 7705
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 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-dom 8921
This theorem is referenced by:  f1dom2g  8945  undom  9039  f1domfi  9164  f1domfi2  9165  sucdom2  9186
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