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Theorem f1imacnv 6266
Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.)
Assertion
Ref Expression
f1imacnv ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem f1imacnv
StepHypRef Expression
1 resima 5541 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 df-f1 6006 . . . . . . 7 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
32simprbi 483 . . . . . 6 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
43adantr 472 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun 𝐹)
5 funcnvres 6080 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
64, 5syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
76imaeq1d 5575 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
8 f1ores 6264 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
9 f1ocnv 6262 . . . . 5 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
108, 9syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
11 imadmrn 5586 . . . . 5 ((𝐹𝐶) “ dom (𝐹𝐶)) = ran (𝐹𝐶)
12 f1odm 6254 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → dom (𝐹𝐶) = (𝐹𝐶))
1312imaeq2d 5576 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ dom (𝐹𝐶)) = ((𝐹𝐶) “ (𝐹𝐶)))
14 f1ofo 6257 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶(𝐹𝐶):(𝐹𝐶)–onto𝐶)
15 forn 6231 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ran (𝐹𝐶) = 𝐶)
1614, 15syl 17 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ran (𝐹𝐶) = 𝐶)
1711, 13, 163eqtr3a 2782 . . . 4 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
1810, 17syl 17 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
197, 18eqtr3d 2760 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
201, 19syl5eqr 2772 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wss 3680  ccnv 5217  dom cdm 5218  ran crn 5219  cres 5220  cima 5221  Fun wfun 5995  wf 5997  1-1wf1 5998  ontowfo 5999  1-1-ontowf1o 6000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008
This theorem is referenced by:  f1opw2  7005  ssenen  8250  f1opwfi  8386  isf34lem3  9310  subggim  17830  gicsubgen  17842  cnt1  21277  basqtop  21637  tgqtop  21638  hmeoopn  21692  hmeocld  21693  hmeontr  21695  qtopf1  21742  f1otrg  25871  tpr2rico  30188  eulerpartlemmf  30667  ballotlemscr  30810  ballotlemrinv0  30824  cvmlift2lem9a  31513  grpokerinj  33924
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