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Theorem f1ocnvfvrneq 5799
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 5487 . . 3 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ocnv 5489 . . 3 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)
3 f1of1 5475 . . 3 (◑𝐹:ran 𝐹–1-1-onto→𝐴 β†’ ◑𝐹:ran 𝐹–1-1→𝐴)
4 f1veqaeq 5786 . . . 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 115 . . 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 18 . 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 124 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  β—‘ccnv 4640  ran crn 4642  β€“1-1β†’wf1 5228  β€“1-1-ontoβ†’wf1o 5230  β€˜cfv 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239
This theorem is referenced by: (None)
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