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Theorem f1imacnv 6865
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 6035 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 df-f1 6568 . . . . . . 7 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
32simprbi 496 . . . . . 6 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
43adantr 480 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun 𝐹)
5 funcnvres 6646 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
64, 5syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
76imaeq1d 6079 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
8 f1ores 6863 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
9 f1ocnv 6861 . . . . 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 6090 . . . . 5 ((𝐹𝐶) “ dom (𝐹𝐶)) = ran (𝐹𝐶)
12 f1odm 6853 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → dom (𝐹𝐶) = (𝐹𝐶))
1312imaeq2d 6080 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ dom (𝐹𝐶)) = ((𝐹𝐶) “ (𝐹𝐶)))
14 f1ofo 6856 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶(𝐹𝐶):(𝐹𝐶)–onto𝐶)
15 forn 6824 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ran (𝐹𝐶) = 𝐶)
1614, 15syl 17 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ran (𝐹𝐶) = 𝐶)
1711, 13, 163eqtr3a 2799 . . . 4 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
1810, 17syl 17 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
197, 18eqtr3d 2777 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
201, 19eqtr3id 2789 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wss 3963  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  f1opw2  7688  mptcnfimad  8010  ssenen  9190  f1opwfi  9394  isf34lem3  10413  subggim  19297  gicsubgen  19310  cnt1  23374  basqtop  23735  tgqtop  23736  hmeoopn  23790  hmeocld  23791  hmeontr  23793  qtopf1  23840  f1otrg  28894  tpr2rico  33873  eulerpartlemmf  34357  ballotlemscr  34500  ballotlemrinv0  34514  cvmlift2lem9a  35288  grpokerinj  37880  uspgrimprop  47811  grimcnv  47817  isubgr3stgrlem8  47876
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