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Theorem f1eq123d 5360
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 5323 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴1-1𝐶𝐺:𝐴1-1𝐶))
31, 2syl 14 . 2 (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐴1-1𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 f1eq2 5324 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴1-1𝐶𝐺:𝐵1-1𝐶))
64, 5syl 14 . 2 (𝜑 → (𝐺:𝐴1-1𝐶𝐺:𝐵1-1𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 f1eq3 5325 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵1-1𝐶𝐺:𝐵1-1𝐷))
97, 8syl 14 . 2 (𝜑 → (𝐺:𝐵1-1𝐶𝐺:𝐵1-1𝐷))
103, 6, 93bitrd 213 1 (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐵1-1𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  1-1wf1 5120
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128
This theorem is referenced by: (None)
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