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Theorem f1imaen2g 9055
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 9054 does not need ax-rep 5279.) (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 6804 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
4 fimass 6756 . . . . . 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 5324 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ∈ V)
8 f1ores 6862 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
98ad2ant2r 747 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
10 f1oen2g 9009 . . 3 ((𝐶𝑉 ∧ (𝐹𝐶) ∈ V ∧ (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶)) → 𝐶 ≈ (𝐹𝐶))
111, 7, 9, 10syl3anc 1373 . 2 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶 ≈ (𝐹𝐶))
1211ensymd 9045 1 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cres 5687  cima 5688  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-er 8745  df-en 8986
This theorem is referenced by:  ssenen  9191  phplem4OLD  9257  fiintOLD  9367  unxpwdom2  9628  znunithash  21583
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