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Theorem f1ocnvfvrneq 7289
Description: If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Assertion
Ref Expression
f1ocnvfvrneq ((𝐹:𝐴–1-1→𝐡 ∧ (𝐢 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) β†’ ((β—‘πΉβ€˜πΆ) = (β—‘πΉβ€˜π·) β†’ 𝐢 = 𝐷))

Proof of Theorem f1ocnvfvrneq
StepHypRef Expression
1 f1f1orn 6844 . . 3 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ocnv 6845 . . 3 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)
3 f1of1 6832 . . 3 (◑𝐹:ran 𝐹–1-1-onto→𝐴 β†’ ◑𝐹:ran 𝐹–1-1→𝐴)
4 f1veqaeq 7261 . . . 4 ((◑𝐹:ran 𝐹–1-1→𝐴 ∧ (𝐢 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) β†’ ((β—‘πΉβ€˜πΆ) = (β—‘πΉβ€˜π·) β†’ 𝐢 = 𝐷))
54ex 412 . . 3 (◑𝐹:ran 𝐹–1-1→𝐴 β†’ ((𝐢 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹) β†’ ((β—‘πΉβ€˜πΆ) = (β—‘πΉβ€˜π·) β†’ 𝐢 = 𝐷)))
61, 2, 3, 54syl 19 . 2 (𝐹:𝐴–1-1→𝐡 β†’ ((𝐢 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹) β†’ ((β—‘πΉβ€˜πΆ) = (β—‘πΉβ€˜π·) β†’ 𝐢 = 𝐷)))
76imp 406 1 ((𝐹:𝐴–1-1→𝐡 ∧ (𝐢 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) β†’ ((β—‘πΉβ€˜πΆ) = (β—‘πΉβ€˜π·) β†’ 𝐢 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  β—‘ccnv 5671  ran crn 5673  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550
This theorem is referenced by: (None)
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