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Theorem f1oen4g 8960
Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8967 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 6822 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 3588 . . . 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 8948 . . . 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 5149  1-1-ontowf1o 6543  cen 8936
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 5300  ax-nul 5307  ax-pr 5428
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-en 8940
This theorem is referenced by:  dif1enlem  9156
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