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Theorem f1imacnv 6801
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 5972 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 df-f1 6502 . . . . . . 7 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
32simprbi 498 . . . . . 6 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
43adantr 482 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun 𝐹)
5 funcnvres 6580 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
64, 5syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
76imaeq1d 6013 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
8 f1ores 6799 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
9 f1ocnv 6797 . . . . 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 6024 . . . . 5 ((𝐹𝐶) “ dom (𝐹𝐶)) = ran (𝐹𝐶)
12 f1odm 6789 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → dom (𝐹𝐶) = (𝐹𝐶))
1312imaeq2d 6014 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ dom (𝐹𝐶)) = ((𝐹𝐶) “ (𝐹𝐶)))
14 f1ofo 6792 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶(𝐹𝐶):(𝐹𝐶)–onto𝐶)
15 forn 6760 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ran (𝐹𝐶) = 𝐶)
1614, 15syl 17 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ran (𝐹𝐶) = 𝐶)
1711, 13, 163eqtr3a 2801 . . . 4 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
1810, 17syl 17 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
197, 18eqtr3d 2779 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
201, 19eqtr3id 2791 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wss 3911  ccnv 5633  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  Fun wfun 6491  wf 6493  1-1wf1 6494  ontowfo 6495  1-1-ontowf1o 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  f1opw2  7609  ssenen  9096  f1opwfi  9301  isf34lem3  10312  subggim  19057  gicsubgen  19069  cnt1  22704  basqtop  23065  tgqtop  23066  hmeoopn  23120  hmeocld  23121  hmeontr  23123  qtopf1  23170  f1otrg  27816  tpr2rico  32496  eulerpartlemmf  32978  ballotlemscr  33121  ballotlemrinv0  33135  cvmlift2lem9a  33900  grpokerinj  36355  isomgrsym  46035
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