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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 6843 . . 3 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ocnv 6844 . . 3 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)
3 f1of1 6831 . . 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 411 . . 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 405 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 394   = wceq 1533   ∈ wcel 2098  β—‘ccnv 5672  ran crn 5674  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by: (None)
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