New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  foimacnv GIF version

Theorem foimacnv 5303
 Description: A reverse version of f1imacnv 5302. (Contributed by Jeffrey Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv ((F:AontoB C B) → (F “ (FC)) = C)

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 5006 . 2 ((F (FC)) “ (FC)) = (F “ (FC))
2 fofun 5270 . . . . . 6 (F:AontoB → Fun F)
32adantr 451 . . . . 5 ((F:AontoB C B) → Fun F)
4 funcnvres2 5167 . . . . 5 (Fun F(F C) = (F (FC)))
53, 4syl 15 . . . 4 ((F:AontoB C B) → (F C) = (F (FC)))
65imaeq1d 4941 . . 3 ((F:AontoB C B) → ((F C) “ (FC)) = ((F (FC)) “ (FC)))
7 resss 4988 . . . . . . . . . . 11 (F C) F
8 cnvss 4885 . . . . . . . . . . 11 ((F C) F(F C) F)
97, 8ax-mp 8 . . . . . . . . . 10 (F C) F
10 cnvcnv 5062 . . . . . . . . . 10 F = F
119, 10sseqtri 3303 . . . . . . . . 9 (F C) F
12 funss 5126 . . . . . . . . 9 ((F C) F → (Fun F → Fun (F C)))
1311, 2, 12mpsyl 59 . . . . . . . 8 (F:AontoB → Fun (F C))
1413adantr 451 . . . . . . 7 ((F:AontoB C B) → Fun (F C))
15 dfima3 4951 . . . . . . . 8 (FC) = ran (F C)
16 dfrn4 4904 . . . . . . . 8 ran (F C) = dom (F C)
1715, 16eqtr2i 2374 . . . . . . 7 dom (F C) = (FC)
1814, 17jctir 524 . . . . . 6 ((F:AontoB C B) → (Fun (F C) dom (F C) = (FC)))
19 df-fn 4790 . . . . . 6 ((F C) Fn (FC) ↔ (Fun (F C) dom (F C) = (FC)))
2018, 19sylibr 203 . . . . 5 ((F:AontoB C B) → (F C) Fn (FC))
21 df-dm 4787 . . . . . 6 dom (F C) = ran (F C)
22 forn 5272 . . . . . . . . . 10 (F:AontoB → ran F = B)
2322sseq2d 3299 . . . . . . . . 9 (F:AontoB → (C ran FC B))
2423biimpar 471 . . . . . . . 8 ((F:AontoB C B) → C ran F)
25 dfrn4 4904 . . . . . . . 8 ran F = dom F
2624, 25syl6sseq 3317 . . . . . . 7 ((F:AontoB C B) → C dom F)
27 ssdmres 4987 . . . . . . 7 (C dom F ↔ dom (F C) = C)
2826, 27sylib 188 . . . . . 6 ((F:AontoB C B) → dom (F C) = C)
2921, 28syl5eqr 2399 . . . . 5 ((F:AontoB C B) → ran (F C) = C)
30 df-fo 4793 . . . . 5 ((F C):(FC)–ontoC ↔ ((F C) Fn (FC) ran (F C) = C))
3120, 29, 30sylanbrc 645 . . . 4 ((F:AontoB C B) → (F C):(FC)–ontoC)
32 foima 5274 . . . 4 ((F C):(FC)–ontoC → ((F C) “ (FC)) = C)
3331, 32syl 15 . . 3 ((F:AontoB C B) → ((F C) “ (FC)) = C)
346, 33eqtr3d 2387 . 2 ((F:AontoB C B) → ((F (FC)) “ (FC)) = C)
351, 34syl5eqr 2399 1 ((F:AontoB C B) → (F “ (FC)) = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3257   “ cima 4722  ◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774  Fun wfun 4775   Fn wfn 4776  –onto→wfo 4779 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator