MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1eq123d Structured version   Visualization version   GIF version

Theorem f1eq123d 6386
Description: Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
f1eq123d.1 (𝜑𝐹 = 𝐺)
f1eq123d.2 (𝜑𝐴 = 𝐵)
f1eq123d.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
f1eq123d (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐵1-1𝐷))

Proof of Theorem f1eq123d
StepHypRef Expression
1 f1eq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 f1eq1 6348 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴1-1𝐶𝐺:𝐴1-1𝐶))
31, 2syl 17 . 2 (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐴1-1𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 f1eq2 6349 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴1-1𝐶𝐺:𝐵1-1𝐶))
64, 5syl 17 . 2 (𝜑 → (𝐺:𝐴1-1𝐶𝐺:𝐵1-1𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 f1eq3 6350 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵1-1𝐶𝐺:𝐵1-1𝐷))
97, 8syl 17 . 2 (𝜑 → (𝐺:𝐵1-1𝐶𝐺:𝐵1-1𝐷))
103, 6, 93bitrd 297 1 (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐵1-1𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1601  1-1wf1 6134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142
This theorem is referenced by:  f10d  6426  fthf1  16966  cofth  16984  istrkgld  25814  istrkg2ld  25815  isushgr  26413  isuspgr  26505  isusgr  26506  isuspgrop  26514  isusgrop  26515  ausgrusgrb  26518  ausgrusgri  26521  usgrstrrepe  26586  uspgr1e  26595  usgrexmpl  26614  usgrres1  26666  usgrexi  26793  uspgr2wlkeq  26997  usgr2trlncl  27116  aciunf1  30032  dimkerim  30445
  Copyright terms: Public domain W3C validator