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Theorem f1imaen 8560
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
Hypothesis
Ref Expression
f1imaen.1 𝐶 ∈ V
Assertion
Ref Expression
f1imaen ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) ≈ 𝐶)

Proof of Theorem f1imaen
StepHypRef Expression
1 f1imaen.1 . 2 𝐶 ∈ V
2 f1imaeng 8558 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴𝐶 ∈ V) → (𝐹𝐶) ≈ 𝐶)
31, 2mp3an3 1443 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) ≈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2107  Vcvv 3500  wss 3940   class class class wbr 5063  cima 5557  1-1wf1 6349  cen 8495
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-er 8279  df-en 8499
This theorem is referenced by:  ssenen  8680  fin4en1  9720  tskinf  10180  tskuni  10194  isercoll  15014  phimullem  16106  odngen  18622  erdsze2lem2  32338
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