Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  foeq123d GIF version

Theorem foeq123d 5370
 Description: Equality deduction for onto 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
foeq123d (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))

Proof of Theorem foeq123d
StepHypRef Expression
1 f1eq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 foeq1 5350 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
31, 2syl 14 . 2 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 foeq2 5351 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
64, 5syl 14 . 2 (𝜑 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 foeq3 5352 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
97, 8syl 14 . 2 (𝜑 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
103, 6, 93bitrd 213 1 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332  –onto→wfo 5130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-opab 3999  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-fun 5134  df-fn 5135  df-fo 5138 This theorem is referenced by:  ctssexmid  7034  ctiunctal  12010  unct  12011
 Copyright terms: Public domain W3C validator