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Theorem f1oen4g 9004
Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9010 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 6837 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 3597 . . . 4 (𝐹𝑉 → (𝐹:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
32imp 406 . . 3 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
433ad2antl1 1184 . 2 (((𝐹𝑉𝐴𝑊𝐵𝑋) ∧ 𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
5 breng 8993 . . . 4 ((𝐴𝑊𝐵𝑋) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
653adant1 1129 . . 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 1776  wcel 2106   class class class wbr 5148  1-1-ontowf1o 6562  cen 8981
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
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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  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-fo 6569  df-f1o 6570  df-en 8985
This theorem is referenced by:  dif1enlem  9195
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