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Theorem f1imaen3g 9055
Description: If a set function is one-to-one, then a subset of its domain is equinumerous to the image of that subset. (This version of f1imaeng 9053 does not need ax-rep 5285 nor ax-pow 5371.) (Contributed by BTernaryTau, 13-Jan-2025.)
Assertion
Ref Expression
f1imaen3g ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → 𝐶 ≈ (𝐹𝐶))

Proof of Theorem f1imaen3g
StepHypRef Expression
1 resexg 6047 . . 3 (𝐹𝑉 → (𝐹𝐶) ∈ V)
213ad2ant3 1134 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → (𝐹𝐶) ∈ V)
3 f1ores 6863 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
433adant3 1131 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
5 f1oen3g 9006 . 2 (((𝐹𝐶) ∈ V ∧ (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶)) → 𝐶 ≈ (𝐹𝐶))
62, 4, 5syl2anc 584 1 ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → 𝐶 ≈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2106  Vcvv 3478  wss 3963   class class class wbr 5148  cres 5691  cima 5692  1-1wf1 6560  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  ax-un 7754
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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  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:  fiint  9364
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