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Theorem f1ocnvfvrneq 7047
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:𝐴1-1𝐵 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1ocnvfvrneq
StepHypRef Expression
1 f1f1orn 6623 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
2 f1ocnv 6624 . . 3 (𝐹:𝐴1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝐴)
3 f1of1 6611 . . 3 (𝐹:ran 𝐹1-1-onto𝐴𝐹:ran 𝐹1-1𝐴)
4 f1veqaeq 7020 . . . 4 ((𝐹:ran 𝐹1-1𝐴 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
54ex 416 . . 3 (𝐹:ran 𝐹1-1𝐴 → ((𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
61, 2, 3, 54syl 19 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
76imp 410 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  ccnv 5518  ran crn 5520  1-1wf1 6330  1-1-ontowf1o 6332  cfv 6333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341
This theorem is referenced by: (None)
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