MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1imacnv Structured version   Visualization version   GIF version

Theorem f1imacnv 6294
Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.)
Assertion
Ref Expression
f1imacnv ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Proof of Theorem f1imacnv
StepHypRef Expression
1 resima 5572 . 2 ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = (𝐹 “ (𝐹𝐶))
2 df-f1 6036 . . . . . . 7 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
32simprbi 484 . . . . . 6 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
43adantr 466 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun 𝐹)
5 funcnvres 6107 . . . . 5 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
64, 5syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
76imaeq1d 5606 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)))
8 f1ores 6292 . . . . 5 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
9 f1ocnv 6290 . . . . 5 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
108, 9syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
11 imadmrn 5617 . . . . 5 ((𝐹𝐶) “ dom (𝐹𝐶)) = ran (𝐹𝐶)
12 f1odm 6282 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → dom (𝐹𝐶) = (𝐹𝐶))
1312imaeq2d 5607 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ dom (𝐹𝐶)) = ((𝐹𝐶) “ (𝐹𝐶)))
14 f1ofo 6285 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶(𝐹𝐶):(𝐹𝐶)–onto𝐶)
15 forn 6259 . . . . . 6 ((𝐹𝐶):(𝐹𝐶)–onto𝐶 → ran (𝐹𝐶) = 𝐶)
1614, 15syl 17 . . . . 5 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ran (𝐹𝐶) = 𝐶)
1711, 13, 163eqtr3a 2829 . . . 4 ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
1810, 17syl 17 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹𝐶) “ (𝐹𝐶)) = 𝐶)
197, 18eqtr3d 2807 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → ((𝐹 ↾ (𝐹𝐶)) “ (𝐹𝐶)) = 𝐶)
201, 19syl5eqr 2819 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wss 3723  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Fun wfun 6025  wf 6027  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038
This theorem is referenced by:  f1opw2  7035  ssenen  8290  f1opwfi  8426  isf34lem3  9399  subggim  17916  gicsubgen  17928  cnt1  21375  basqtop  21735  tgqtop  21736  hmeoopn  21790  hmeocld  21791  hmeontr  21793  qtopf1  21840  f1otrg  25972  tpr2rico  30298  eulerpartlemmf  30777  ballotlemscr  30920  ballotlemrinv0  30934  cvmlift2lem9a  31623  grpokerinj  34024
  Copyright terms: Public domain W3C validator