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Theorem foimacnv 6625
Description: A reverse version of f1imacnv 6624. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 5880 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 fofun 6584 . . . . . 6 (𝐹:𝐴onto𝐵 → Fun 𝐹)
32adantr 481 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun 𝐹)
4 funcnvres2 6427 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
53, 4syl 17 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
65imaeq1d 5921 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
7 resss 5871 . . . . . . . . . 10 (𝐹𝐶) ⊆ 𝐹
8 cnvss 5736 . . . . . . . . . 10 ((𝐹𝐶) ⊆ 𝐹(𝐹𝐶) ⊆ 𝐹)
97, 8ax-mp 5 . . . . . . . . 9 (𝐹𝐶) ⊆ 𝐹
10 cnvcnvss 6044 . . . . . . . . 9 𝐹𝐹
119, 10sstri 3973 . . . . . . . 8 (𝐹𝐶) ⊆ 𝐹
12 funss 6367 . . . . . . . 8 ((𝐹𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐶)))
1311, 2, 12mpsyl 68 . . . . . . 7 (𝐹:𝐴onto𝐵 → Fun (𝐹𝐶))
1413adantr 481 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun (𝐹𝐶))
15 df-ima 5561 . . . . . . 7 (𝐹𝐶) = ran (𝐹𝐶)
16 df-rn 5559 . . . . . . 7 ran (𝐹𝐶) = dom (𝐹𝐶)
1715, 16eqtr2i 2842 . . . . . 6 dom (𝐹𝐶) = (𝐹𝐶)
18 df-fn 6351 . . . . . 6 ((𝐹𝐶) Fn (𝐹𝐶) ↔ (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
1914, 17, 18sylanblrc 590 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) Fn (𝐹𝐶))
20 dfdm4 5757 . . . . . 6 dom (𝐹𝐶) = ran (𝐹𝐶)
21 forn 6586 . . . . . . . . . 10 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
2221sseq2d 3996 . . . . . . . . 9 (𝐹:𝐴onto𝐵 → (𝐶 ⊆ ran 𝐹𝐶𝐵))
2322biimpar 478 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ ran 𝐹)
24 df-rn 5559 . . . . . . . 8 ran 𝐹 = dom 𝐹
2523, 24sseqtrdi 4014 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ dom 𝐹)
26 ssdmres 5869 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
2725, 26sylib 219 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → dom (𝐹𝐶) = 𝐶)
2820, 27syl5eqr 2867 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → ran (𝐹𝐶) = 𝐶)
29 df-fo 6354 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 ↔ ((𝐹𝐶) Fn (𝐹𝐶) ∧ ran (𝐹𝐶) = 𝐶))
3019, 28, 29sylanbrc 583 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶):(𝐹𝐶)–onto𝐶)
31 foima 6588 . . . 4 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
3230, 31syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
336, 32eqtr3d 2855 . 2 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
341, 33syl5eqr 2867 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wss 3933  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Fun wfun 6342   Fn wfn 6343  ontowfo 6346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354
This theorem is referenced by:  f1opw2  7389  imacosupp  7863  imacosuppOLD  7864  fopwdom  8613  f1opwfi  8816  enfin2i  9731  fin1a2lem7  9816  fsumss  15070  fprodss  15290  gicsubgen  18356  coe1mul2lem2  20364  cncmp  21928  cnconn  21958  qtoprest  22253  qtopomap  22254  qtopcmap  22255  hmeoimaf1o  22306  elfm3  22486  imasf1oxms  23026  mbfimaopnlem  24183  cvmsss2  32418  diaintclN  38074  dibintclN  38183  dihintcl  38360  lnmepi  39563  pwfi2f1o  39574  sge0f1o  42541
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