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Theorem f1oen4g 9006
Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9012 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 6835 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 3596 . . . 4 (𝐹𝑉 → (𝐹:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
32imp 406 . . 3 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
433ad2antl1 1185 . 2 (((𝐹𝑉𝐴𝑊𝐵𝑋) ∧ 𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
5 breng 8995 . . . 4 ((𝐴𝑊𝐵𝑋) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
653adant1 1130 . . 3 ((𝐹𝑉𝐴𝑊𝐵𝑋) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
76adantr 480 . 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 206  wa 395  w3a 1086  wex 1778  wcel 2107   class class class wbr 5142  1-1-ontowf1o 6559  cen 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-en 8987
This theorem is referenced by:  dif1enlem  9197
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