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Theorem f1ocnvfvrneq 5615
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 5312 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
2 f1ocnv 5314 . . 3 (𝐹:𝐴1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝐴)
3 f1of1 5300 . . 3 (𝐹:ran 𝐹1-1-onto𝐴𝐹:ran 𝐹1-1𝐴)
4 f1veqaeq 5602 . . . 4 ((𝐹:ran 𝐹1-1𝐴 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
54ex 114 . . 3 (𝐹:ran 𝐹1-1𝐴 → ((𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
61, 2, 3, 54syl 18 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
76imp 123 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1299  wcel 1448  ccnv 4476  ran crn 4478  1-1wf1 5056  1-1-ontowf1o 5058  cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067
This theorem is referenced by: (None)
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