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Theorem foimacnv 5474
Description: A reverse version of f1imacnv 5473. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 4935 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 fofun 5434 . . . . . 6 (𝐹:𝐴onto𝐵 → Fun 𝐹)
32adantr 276 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun 𝐹)
4 funcnvres2 5286 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
53, 4syl 14 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
65imaeq1d 4964 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
7 resss 4926 . . . . . . . . . . 11 (𝐹𝐶) ⊆ 𝐹
8 cnvss 4795 . . . . . . . . . . 11 ((𝐹𝐶) ⊆ 𝐹(𝐹𝐶) ⊆ 𝐹)
97, 8ax-mp 5 . . . . . . . . . 10 (𝐹𝐶) ⊆ 𝐹
10 cnvcnvss 5078 . . . . . . . . . 10 𝐹𝐹
119, 10sstri 3164 . . . . . . . . 9 (𝐹𝐶) ⊆ 𝐹
12 funss 5230 . . . . . . . . 9 ((𝐹𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐶)))
1311, 2, 12mpsyl 65 . . . . . . . 8 (𝐹:𝐴onto𝐵 → Fun (𝐹𝐶))
1413adantr 276 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun (𝐹𝐶))
15 df-ima 4635 . . . . . . . 8 (𝐹𝐶) = ran (𝐹𝐶)
16 df-rn 4633 . . . . . . . 8 ran (𝐹𝐶) = dom (𝐹𝐶)
1715, 16eqtr2i 2199 . . . . . . 7 dom (𝐹𝐶) = (𝐹𝐶)
1814, 17jctir 313 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
19 df-fn 5214 . . . . . 6 ((𝐹𝐶) Fn (𝐹𝐶) ↔ (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
2018, 19sylibr 134 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) Fn (𝐹𝐶))
21 dfdm4 4814 . . . . . 6 dom (𝐹𝐶) = ran (𝐹𝐶)
22 forn 5436 . . . . . . . . . 10 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
2322sseq2d 3185 . . . . . . . . 9 (𝐹:𝐴onto𝐵 → (𝐶 ⊆ ran 𝐹𝐶𝐵))
2423biimpar 297 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ ran 𝐹)
25 df-rn 4633 . . . . . . . 8 ran 𝐹 = dom 𝐹
2624, 25sseqtrdi 3203 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ dom 𝐹)
27 ssdmres 4924 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
2826, 27sylib 122 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → dom (𝐹𝐶) = 𝐶)
2921, 28eqtr3id 2224 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → ran (𝐹𝐶) = 𝐶)
30 df-fo 5217 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 ↔ ((𝐹𝐶) Fn (𝐹𝐶) ∧ ran (𝐹𝐶) = 𝐶))
3120, 29, 30sylanbrc 417 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶):(𝐹𝐶)–onto𝐶)
32 foima 5438 . . . 4 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
3331, 32syl 14 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
346, 33eqtr3d 2212 . 2 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
351, 34eqtr3id 2224 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wss 3129  ccnv 4621  dom cdm 4622  ran crn 4623  cres 4624  cima 4625  Fun wfun 5205   Fn wfn 5206  ontowfo 5209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-fun 5213  df-fn 5214  df-f 5215  df-fo 5217
This theorem is referenced by:  f1opw2  6070  fopwdom  6829  fisumss  11371  fprodssdc  11569  hmeoimaf1o  13447
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