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Theorem f1imaen2g 9044
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 9043 does not need ax-rep 5289.) (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 771 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶𝑉)
2 simplr 767 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐵𝑉)
3 f1f 6798 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
4 fimass 6748 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐶) ⊆ 𝐵)
53, 4syl 17 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹𝐶) ⊆ 𝐵)
65ad2antrr 724 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ⊆ 𝐵)
72, 6ssexd 5328 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ∈ V)
8 f1ores 6858 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
98ad2ant2r 745 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
10 f1oen2g 8997 . . 3 ((𝐶𝑉 ∧ (𝐹𝐶) ∈ V ∧ (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶)) → 𝐶 ≈ (𝐹𝐶))
111, 7, 9, 10syl3anc 1368 . 2 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶 ≈ (𝐹𝐶))
1211ensymd 9034 1 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  Vcvv 3473  wss 3949   class class class wbr 5152  cres 5684  cima 5685  wf 6549  1-1wf1 6550  1-1-ontowf1o 6552  cen 8969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-er 8733  df-en 8973
This theorem is referenced by:  ssenen  9184  phplem4OLD  9253  fiint  9358  unxpwdom2  9621  znunithash  21512
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