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Theorem f1imacnv 5302
 Description: Preimage of an image. (Contributed by set.mm contributors, 30-Sep-2004.)
Assertion
Ref Expression
f1imacnv ((F:A1-1B C A) → (F “ (FC)) = C)

Proof of Theorem f1imacnv
StepHypRef Expression
1 resima 5006 . 2 ((F (FC)) “ (FC)) = (F “ (FC))
2 df-f1 4792 . . . . . 6 (F:A1-1B ↔ (F:A–→B Fun F))
32simprbi 450 . . . . 5 (F:A1-1B → Fun F)
43adantr 451 . . . 4 ((F:A1-1B C A) → Fun F)
5 funcnvres 5165 . . . 4 (Fun F(F C) = (F (FC)))
6 imaeq1 4937 . . . 4 ((F C) = (F (FC)) → ((F C) “ (FC)) = ((F (FC)) “ (FC)))
74, 5, 63syl 18 . . 3 ((F:A1-1B C A) → ((F C) “ (FC)) = ((F (FC)) “ (FC)))
8 f1ores 5300 . . . 4 ((F:A1-1B C A) → (F C):C1-1-onto→(FC))
9 f1ocnv 5299 . . . 4 ((F C):C1-1-onto→(FC) → (F C):(FC)–1-1-ontoC)
10 imadmrn 5008 . . . . . 6 ((F C) “ dom (F C)) = ran (F C)
11 f1of 5287 . . . . . . 7 ((F C):(FC)–1-1-ontoC(F C):(FC)–→C)
12 fdm 5226 . . . . . . 7 ((F C):(FC)–→C → dom (F C) = (FC))
13 imaeq2 4938 . . . . . . 7 (dom (F C) = (FC) → ((F C) “ dom (F C)) = ((F C) “ (FC)))
1411, 12, 133syl 18 . . . . . 6 ((F C):(FC)–1-1-ontoC → ((F C) “ dom (F C)) = ((F C) “ (FC)))
1510, 14syl5reqr 2400 . . . . 5 ((F C):(FC)–1-1-ontoC → ((F C) “ (FC)) = ran (F C))
16 f1ofo 5293 . . . . . 6 ((F C):(FC)–1-1-ontoC(F C):(FC)–ontoC)
17 forn 5272 . . . . . 6 ((F C):(FC)–ontoC → ran (F C) = C)
1816, 17syl 15 . . . . 5 ((F C):(FC)–1-1-ontoC → ran (F C) = C)
1915, 18eqtrd 2385 . . . 4 ((F C):(FC)–1-1-ontoC → ((F C) “ (FC)) = C)
208, 9, 193syl 18 . . 3 ((F:A1-1B C A) → ((F C) “ (FC)) = C)
217, 20eqtr3d 2387 . 2 ((F:A1-1B C A) → ((F (FC)) “ (FC)) = C)
221, 21syl5eqr 2399 1 ((F:A1-1B C A) → (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  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780 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-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by: (None)
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