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Theorem f1ocnvfvrneq 7233
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 6796 . . 3 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ocnv 6797 . . 3 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)
3 f1of1 6784 . . 3 (◑𝐹:ran 𝐹–1-1-onto→𝐴 β†’ ◑𝐹:ran 𝐹–1-1→𝐴)
4 f1veqaeq 7205 . . . 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 414 . . 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 408 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 397   = wceq 1542   ∈ wcel 2107  β—‘ccnv 5633  ran crn 5635  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by: (None)
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