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Theorem f1imaen2g 9054
Description: If a function is one-to-one, then the image of a subset of its domain under it is equinumerous to the subset. (This version of f1imaeng 9053 does not need ax-rep 5285.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
f1imaen2g (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)

Proof of Theorem f1imaen2g
StepHypRef Expression
1 simprr 773 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶𝑉)
2 simplr 769 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐵𝑉)
3 f1f 6805 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
4 fimass 6757 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐶) ⊆ 𝐵)
53, 4syl 17 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹𝐶) ⊆ 𝐵)
65ad2antrr 726 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ⊆ 𝐵)
72, 6ssexd 5330 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ∈ V)
8 f1ores 6863 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
98ad2ant2r 747 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
10 f1oen2g 9008 . . 3 ((𝐶𝑉 ∧ (𝐹𝐶) ∈ V ∧ (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶)) → 𝐶 ≈ (𝐹𝐶))
111, 7, 9, 10syl3anc 1370 . 2 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶 ≈ (𝐹𝐶))
1211ensymd 9044 1 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  wss 3963   class class class wbr 5148  cres 5691  cima 5692  wf 6559  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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  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-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  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-er 8744  df-en 8985
This theorem is referenced by:  ssenen  9190  phplem4OLD  9255  fiintOLD  9365  unxpwdom2  9626  znunithash  21601
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