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Theorem f1imacnv 5086
Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.)
Assertion
Ref Expression
f1imacnv ((𝐹:A1-1B 𝐶A) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem f1imacnv
StepHypRef Expression
1 resima 4586 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 df-f1 4850 . . . . . . 7 (𝐹:A1-1B ↔ (𝐹:AB Fun 𝐹))
32simprbi 260 . . . . . 6 (𝐹:A1-1B → Fun 𝐹)
43adantr 261 . . . . 5 ((𝐹:A1-1B 𝐶A) → Fun 𝐹)
5 funcnvres 4915 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
64, 5syl 14 . . . 4 ((𝐹:A1-1B 𝐶A) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
76imaeq1d 4610 . . 3 ((𝐹:A1-1B 𝐶A) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
8 f1ores 5084 . . . . 5 ((𝐹:A1-1B 𝐶A) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
9 f1ocnv 5082 . . . . 5 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
108, 9syl 14 . . . 4 ((𝐹:A1-1B 𝐶A) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
11 imadmrn 4621 . . . . 5 ((𝐹𝐶) “ dom (𝐹𝐶)) = ran (𝐹𝐶)
12 f1odm 5073 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → dom (𝐹𝐶) = (𝐹𝐶))
1312imaeq2d 4611 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ dom (𝐹𝐶)) = ((𝐹𝐶) “ (𝐹𝐶)))
14 f1ofo 5076 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶(𝐹𝐶):(𝐹𝐶)–onto𝐶)
15 forn 5052 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ran (𝐹𝐶) = 𝐶)
1614, 15syl 14 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ran (𝐹𝐶) = 𝐶)
1711, 13, 163eqtr3a 2093 . . . 4 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
1810, 17syl 14 . . 3 ((𝐹:A1-1B 𝐶A) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
197, 18eqtr3d 2071 . 2 ((𝐹:A1-1B 𝐶A) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
201, 19syl5eqr 2083 1 ((𝐹:A1-1B 𝐶A) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wss 2911  ccnv 4287  dom cdm 4288  ran crn 4289  cres 4290  cima 4291  Fun wfun 4839  wf 4841  1-1wf1 4842  ontowfo 4843  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1opw2  5648
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