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Theorem foimacnv 6604
Description: A reverse version of f1imacnv 6603. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 5859 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 fofun 6563 . . . . . 6 (𝐹:𝐴onto𝐵 → Fun 𝐹)
32adantr 483 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun 𝐹)
4 funcnvres2 6406 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
53, 4syl 17 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
65imaeq1d 5900 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
7 resss 5850 . . . . . . . . . 10 (𝐹𝐶) ⊆ 𝐹
8 cnvss 5715 . . . . . . . . . 10 ((𝐹𝐶) ⊆ 𝐹(𝐹𝐶) ⊆ 𝐹)
97, 8ax-mp 5 . . . . . . . . 9 (𝐹𝐶) ⊆ 𝐹
10 cnvcnvss 6023 . . . . . . . . 9 𝐹𝐹
119, 10sstri 3951 . . . . . . . 8 (𝐹𝐶) ⊆ 𝐹
12 funss 6346 . . . . . . . 8 ((𝐹𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐶)))
1311, 2, 12mpsyl 68 . . . . . . 7 (𝐹:𝐴onto𝐵 → Fun (𝐹𝐶))
1413adantr 483 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun (𝐹𝐶))
15 df-ima 5540 . . . . . . 7 (𝐹𝐶) = ran (𝐹𝐶)
16 df-rn 5538 . . . . . . 7 ran (𝐹𝐶) = dom (𝐹𝐶)
1715, 16eqtr2i 2844 . . . . . 6 dom (𝐹𝐶) = (𝐹𝐶)
18 df-fn 6330 . . . . . 6 ((𝐹𝐶) Fn (𝐹𝐶) ↔ (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
1914, 17, 18sylanblrc 592 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) Fn (𝐹𝐶))
20 dfdm4 5736 . . . . . 6 dom (𝐹𝐶) = ran (𝐹𝐶)
21 forn 6565 . . . . . . . . . 10 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
2221sseq2d 3974 . . . . . . . . 9 (𝐹:𝐴onto𝐵 → (𝐶 ⊆ ran 𝐹𝐶𝐵))
2322biimpar 480 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ ran 𝐹)
24 df-rn 5538 . . . . . . . 8 ran 𝐹 = dom 𝐹
2523, 24sseqtrdi 3992 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ dom 𝐹)
26 ssdmres 5848 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
2725, 26sylib 220 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → dom (𝐹𝐶) = 𝐶)
2820, 27syl5eqr 2869 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → ran (𝐹𝐶) = 𝐶)
29 df-fo 6333 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 ↔ ((𝐹𝐶) Fn (𝐹𝐶) ∧ ran (𝐹𝐶) = 𝐶))
3019, 28, 29sylanbrc 585 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶):(𝐹𝐶)–onto𝐶)
31 foima 6567 . . . 4 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
3230, 31syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
336, 32eqtr3d 2857 . 2 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
341, 33syl5eqr 2869 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wss 3909  ccnv 5526  dom cdm 5527  ran crn 5528  cres 5529  cima 5530  Fun wfun 6321   Fn wfn 6322  ontowfo 6325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5039  df-opab 5101  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-fun 6329  df-fn 6330  df-f 6331  df-fo 6333
This theorem is referenced by:  f1opw2  7374  imacosupp  7848  imacosuppOLD  7849  fopwdom  8599  f1opwfi  8802  enfin2i  9717  fin1a2lem7  9802  fsumss  15058  fprodss  15278  gicsubgen  18393  coe1mul2lem2  20408  cncmp  21972  cnconn  22002  qtoprest  22297  qtopomap  22298  qtopcmap  22299  hmeoimaf1o  22350  elfm3  22530  imasf1oxms  23071  mbfimaopnlem  24234  cvmsss2  32525  diaintclN  38226  dibintclN  38335  dihintcl  38512  lnmepi  39818  pwfi2f1o  39829  sge0f1o  42808
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