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Theorem foimacnv 6067
Description: A reverse version of f1imacnv 6066. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 5351 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 fofun 6029 . . . . . 6 (𝐹:𝐴onto𝐵 → Fun 𝐹)
32adantr 480 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun 𝐹)
4 funcnvres2 5883 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
53, 4syl 17 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
65imaeq1d 5384 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
7 resss 5342 . . . . . . . . . . 11 (𝐹𝐶) ⊆ 𝐹
8 cnvss 5216 . . . . . . . . . . 11 ((𝐹𝐶) ⊆ 𝐹(𝐹𝐶) ⊆ 𝐹)
97, 8ax-mp 5 . . . . . . . . . 10 (𝐹𝐶) ⊆ 𝐹
10 cnvcnvss 5507 . . . . . . . . . 10 𝐹𝐹
119, 10sstri 3577 . . . . . . . . 9 (𝐹𝐶) ⊆ 𝐹
12 funss 5822 . . . . . . . . 9 ((𝐹𝐶) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐶)))
1311, 2, 12mpsyl 66 . . . . . . . 8 (𝐹:𝐴onto𝐵 → Fun (𝐹𝐶))
1413adantr 480 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → Fun (𝐹𝐶))
15 df-ima 5051 . . . . . . . 8 (𝐹𝐶) = ran (𝐹𝐶)
16 df-rn 5049 . . . . . . . 8 ran (𝐹𝐶) = dom (𝐹𝐶)
1715, 16eqtr2i 2633 . . . . . . 7 dom (𝐹𝐶) = (𝐹𝐶)
1814, 17jctir 559 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
19 df-fn 5807 . . . . . 6 ((𝐹𝐶) Fn (𝐹𝐶) ↔ (Fun (𝐹𝐶) ∧ dom (𝐹𝐶) = (𝐹𝐶)))
2018, 19sylibr 223 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶) Fn (𝐹𝐶))
21 dfdm4 5238 . . . . . 6 dom (𝐹𝐶) = ran (𝐹𝐶)
22 forn 6031 . . . . . . . . . 10 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
2322sseq2d 3596 . . . . . . . . 9 (𝐹:𝐴onto𝐵 → (𝐶 ⊆ ran 𝐹𝐶𝐵))
2423biimpar 501 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ ran 𝐹)
25 df-rn 5049 . . . . . . . 8 ran 𝐹 = dom 𝐹
2624, 25syl6sseq 3614 . . . . . . 7 ((𝐹:𝐴onto𝐵𝐶𝐵) → 𝐶 ⊆ dom 𝐹)
27 ssdmres 5340 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
2826, 27sylib 207 . . . . . 6 ((𝐹:𝐴onto𝐵𝐶𝐵) → dom (𝐹𝐶) = 𝐶)
2921, 28syl5eqr 2658 . . . . 5 ((𝐹:𝐴onto𝐵𝐶𝐵) → ran (𝐹𝐶) = 𝐶)
30 df-fo 5810 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 ↔ ((𝐹𝐶) Fn (𝐹𝐶) ∧ ran (𝐹𝐶) = 𝐶))
3120, 29, 30sylanbrc 695 . . . 4 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹𝐶):(𝐹𝐶)–onto𝐶)
32 foima 6033 . . . 4 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
3331, 32syl 17 . . 3 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
346, 33eqtr3d 2646 . 2 ((𝐹:𝐴onto𝐵𝐶𝐵) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
351, 34syl5eqr 2658 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wss 3540  ccnv 5037  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Fun wfun 5798   Fn wfn 5799  ontowfo 5802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810
This theorem is referenced by:  f1opw2  6786  imacosupp  7222  fopwdom  7953  f1opwfi  8153  enfin2i  9026  fin1a2lem7  9111  fsumss  14303  fprodss  14517  gicsubgen  17544  coe1mul2lem2  19459  cncmp  21005  cnconn  21035  qtoprest  21330  qtopomap  21331  qtopcmap  21332  hmeoimaf1o  21383  elfm3  21564  imasf1oxms  22104  mbfimaopnlem  23228  cvmsss2  30510  diaintclN  35365  dibintclN  35474  dihintcl  35651  lnmepi  36673  pwfi2f1o  36684  sge0f1o  39275
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