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Theorem foimacnv 5394
 Description: A reverse version of f1imacnv 5393. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 4861 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 fofun 5355 . . . . . 6 (𝐹:𝐴onto𝐵 → Fun 𝐹)
32adantr 274 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun 𝐹)
4 funcnvres2 5207 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
53, 4syl 14 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
65imaeq1d 4889 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
7 resss 4852 . . . . . . . . . . 11 (𝐹𝐶) ⊆ 𝐹
8 cnvss 4721 . . . . . . . . . . 11 ((𝐹𝐶) ⊆ 𝐹(𝐹𝐶) ⊆ 𝐹)
97, 8ax-mp 5 . . . . . . . . . 10 (𝐹𝐶) ⊆ 𝐹
10 cnvcnvss 5002 . . . . . . . . . 10 𝐹𝐹
119, 10sstri 3112 . . . . . . . . 9 (𝐹𝐶) ⊆ 𝐹
12 funss 5151 . . . . . . . . 9 ((𝐹𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐶)))
1311, 2, 12mpsyl 65 . . . . . . . 8 (𝐹:𝐴onto𝐵 → Fun (𝐹𝐶))
1413adantr 274 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun (𝐹𝐶))
15 df-ima 4561 . . . . . . . 8 (𝐹𝐶) = ran (𝐹𝐶)
16 df-rn 4559 . . . . . . . 8 ran (𝐹𝐶) = dom (𝐹𝐶)
1715, 16eqtr2i 2162 . . . . . . 7 dom (𝐹𝐶) = (𝐹𝐶)
1814, 17jctir 311 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
19 df-fn 5135 . . . . . 6 ((𝐹𝐶) Fn (𝐹𝐶) ↔ (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
2018, 19sylibr 133 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) Fn (𝐹𝐶))
21 dfdm4 4740 . . . . . 6 dom (𝐹𝐶) = ran (𝐹𝐶)
22 forn 5357 . . . . . . . . . 10 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
2322sseq2d 3133 . . . . . . . . 9 (𝐹:𝐴onto𝐵 → (𝐶 ⊆ ran 𝐹𝐶𝐵))
2423biimpar 295 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ ran 𝐹)
25 df-rn 4559 . . . . . . . 8 ran 𝐹 = dom 𝐹
2624, 25sseqtrdi 3151 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ dom 𝐹)
27 ssdmres 4850 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
2826, 27sylib 121 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → dom (𝐹𝐶) = 𝐶)
2921, 28syl5eqr 2187 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → ran (𝐹𝐶) = 𝐶)
30 df-fo 5138 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 ↔ ((𝐹𝐶) Fn (𝐹𝐶) ∧ ran (𝐹𝐶) = 𝐶))
3120, 29, 30sylanbrc 414 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶):(𝐹𝐶)–onto𝐶)
32 foima 5359 . . . 4 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
3331, 32syl 14 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
346, 33eqtr3d 2175 . 2 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
351, 34syl5eqr 2187 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ⊆ wss 3077  ◡ccnv 4547  dom cdm 4548  ran crn 4549   ↾ cres 4550   “ cima 4551  Fun wfun 5126   Fn wfn 5127  –onto→wfo 5130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-opab 3999  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-fun 5134  df-fn 5135  df-f 5136  df-fo 5138 This theorem is referenced by:  f1opw2  5985  fopwdom  6739  fisumss  11213  hmeoimaf1o  12542
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