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Theorem foimacnv 6849
Description: A reverse version of f1imacnv 6848. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 6014 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 fofun 6805 . . . . . 6 (𝐹:𝐴onto𝐵 → Fun 𝐹)
32adantr 479 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun 𝐹)
4 funcnvres2 6627 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
53, 4syl 17 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
65imaeq1d 6057 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
7 resss 6005 . . . . . . . . . 10 (𝐹𝐶) ⊆ 𝐹
8 cnvss 5871 . . . . . . . . . 10 ((𝐹𝐶) ⊆ 𝐹(𝐹𝐶) ⊆ 𝐹)
97, 8ax-mp 5 . . . . . . . . 9 (𝐹𝐶) ⊆ 𝐹
10 cnvcnvss 6192 . . . . . . . . 9 𝐹𝐹
119, 10sstri 3990 . . . . . . . 8 (𝐹𝐶) ⊆ 𝐹
12 funss 6566 . . . . . . . 8 ((𝐹𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐶)))
1311, 2, 12mpsyl 68 . . . . . . 7 (𝐹:𝐴onto𝐵 → Fun (𝐹𝐶))
1413adantr 479 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun (𝐹𝐶))
15 df-ima 5688 . . . . . . 7 (𝐹𝐶) = ran (𝐹𝐶)
16 df-rn 5686 . . . . . . 7 ran (𝐹𝐶) = dom (𝐹𝐶)
1715, 16eqtr2i 2759 . . . . . 6 dom (𝐹𝐶) = (𝐹𝐶)
18 df-fn 6545 . . . . . 6 ((𝐹𝐶) Fn (𝐹𝐶) ↔ (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
1914, 17, 18sylanblrc 588 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) Fn (𝐹𝐶))
20 dfdm4 5894 . . . . . 6 dom (𝐹𝐶) = ran (𝐹𝐶)
21 forn 6807 . . . . . . . . . 10 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
2221sseq2d 4013 . . . . . . . . 9 (𝐹:𝐴onto𝐵 → (𝐶 ⊆ ran 𝐹𝐶𝐵))
2322biimpar 476 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ ran 𝐹)
24 df-rn 5686 . . . . . . . 8 ran 𝐹 = dom 𝐹
2523, 24sseqtrdi 4031 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ dom 𝐹)
26 ssdmres 6003 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
2725, 26sylib 217 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → dom (𝐹𝐶) = 𝐶)
2820, 27eqtr3id 2784 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → ran (𝐹𝐶) = 𝐶)
29 df-fo 6548 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 ↔ ((𝐹𝐶) Fn (𝐹𝐶) ∧ ran (𝐹𝐶) = 𝐶))
3019, 28, 29sylanbrc 581 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶):(𝐹𝐶)–onto𝐶)
31 foima 6809 . . . 4 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
3230, 31syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
336, 32eqtr3d 2772 . 2 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
341, 33eqtr3id 2784 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wss 3947  ccnv 5674  dom cdm 5675  ran crn 5676  cres 5677  cima 5678  Fun wfun 6536   Fn wfn 6537  ontowfo 6540
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548
This theorem is referenced by:  f1opw2  7663  imacosupp  8196  fopwdom  9082  f1opwfi  9358  enfin2i  10318  fin1a2lem7  10403  fsumss  15675  fprodss  15896  gicsubgen  19193  coe1mul2lem2  22010  cncmp  23116  cnconn  23146  qtoprest  23441  qtopomap  23442  qtopcmap  23443  hmeoimaf1o  23494  elfm3  23674  imasf1oxms  24218  mbfimaopnlem  25404  cvmsss2  34563  diaintclN  40232  dibintclN  40341  dihintcl  40518  lnmepi  42129  pwfi2f1o  42140  sge0f1o  45396
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