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Theorem foimacnv 6865
Description: A reverse version of f1imacnv 6864. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 6033 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 fofun 6821 . . . . . 6 (𝐹:𝐴onto𝐵 → Fun 𝐹)
32adantr 480 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun 𝐹)
4 funcnvres2 6646 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
53, 4syl 17 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
65imaeq1d 6077 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
7 resss 6019 . . . . . . . . . 10 (𝐹𝐶) ⊆ 𝐹
8 cnvss 5883 . . . . . . . . . 10 ((𝐹𝐶) ⊆ 𝐹(𝐹𝐶) ⊆ 𝐹)
97, 8ax-mp 5 . . . . . . . . 9 (𝐹𝐶) ⊆ 𝐹
10 cnvcnvss 6214 . . . . . . . . 9 𝐹𝐹
119, 10sstri 3993 . . . . . . . 8 (𝐹𝐶) ⊆ 𝐹
12 funss 6585 . . . . . . . 8 ((𝐹𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐶)))
1311, 2, 12mpsyl 68 . . . . . . 7 (𝐹:𝐴onto𝐵 → Fun (𝐹𝐶))
1413adantr 480 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun (𝐹𝐶))
15 df-ima 5698 . . . . . . 7 (𝐹𝐶) = ran (𝐹𝐶)
16 df-rn 5696 . . . . . . 7 ran (𝐹𝐶) = dom (𝐹𝐶)
1715, 16eqtr2i 2766 . . . . . 6 dom (𝐹𝐶) = (𝐹𝐶)
18 df-fn 6564 . . . . . 6 ((𝐹𝐶) Fn (𝐹𝐶) ↔ (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
1914, 17, 18sylanblrc 590 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) Fn (𝐹𝐶))
20 dfdm4 5906 . . . . . 6 dom (𝐹𝐶) = ran (𝐹𝐶)
21 forn 6823 . . . . . . . . . 10 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
2221sseq2d 4016 . . . . . . . . 9 (𝐹:𝐴onto𝐵 → (𝐶 ⊆ ran 𝐹𝐶𝐵))
2322biimpar 477 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ ran 𝐹)
24 df-rn 5696 . . . . . . . 8 ran 𝐹 = dom 𝐹
2523, 24sseqtrdi 4024 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ dom 𝐹)
26 ssdmres 6031 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
2725, 26sylib 218 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → dom (𝐹𝐶) = 𝐶)
2820, 27eqtr3id 2791 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → ran (𝐹𝐶) = 𝐶)
29 df-fo 6567 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 ↔ ((𝐹𝐶) Fn (𝐹𝐶) ∧ ran (𝐹𝐶) = 𝐶))
3019, 28, 29sylanbrc 583 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶):(𝐹𝐶)–onto𝐶)
31 foima 6825 . . . 4 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
3230, 31syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
336, 32eqtr3d 2779 . 2 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
341, 33eqtr3id 2791 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wss 3951  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555   Fn wfn 6556  ontowfo 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567
This theorem is referenced by:  f1opw2  7688  mptcnfimad  8011  imacosupp  8234  fopwdom  9120  f1opwfi  9396  enfin2i  10361  fin1a2lem7  10446  fsumss  15761  fprodss  15984  gicsubgen  19297  coe1mul2lem2  22271  cncmp  23400  cnconn  23430  qtoprest  23725  qtopomap  23726  qtopcmap  23727  hmeoimaf1o  23778  elfm3  23958  imasf1oxms  24502  mbfimaopnlem  25690  cvmsss2  35279  diaintclN  41060  dibintclN  41169  dihintcl  41346  lnmepi  43097  pwfi2f1o  43108  sge0f1o  46397  isubgr3stgrlem8  47940
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