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Theorem f1oen4g 8956
Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8963 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
Assertion
Ref Expression
f1oen4g (((𝐹𝑉𝐴𝑊𝐵𝑋) ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Proof of Theorem f1oen4g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1oeq1 6818 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 3587 . . . 4 (𝐹𝑉 → (𝐹:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
32imp 408 . . 3 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
433ad2antl1 1186 . 2 (((𝐹𝑉𝐴𝑊𝐵𝑋) ∧ 𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
5 breng 8944 . . . 4 ((𝐴𝑊𝐵𝑋) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
653adant1 1131 . . 3 ((𝐹𝑉𝐴𝑊𝐵𝑋) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
76adantr 482 . 2 (((𝐹𝑉𝐴𝑊𝐵𝑋) ∧ 𝐹:𝐴1-1-onto𝐵) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
84, 7mpbird 257 1 (((𝐹𝑉𝐴𝑊𝐵𝑋) ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wex 1782  wcel 2107   class class class wbr 5147  1-1-ontowf1o 6539  cen 8932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-en 8936
This theorem is referenced by:  dif1enlem  9152
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