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Mirrors > Home > ILE Home > Th. List > f1ocnvfvrneq | GIF version |
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.) |
Ref | Expression |
---|---|
f1ocnvfvrneq | β’ ((πΉ:π΄β1-1βπ΅ β§ (πΆ β ran πΉ β§ π· β ran πΉ)) β ((β‘πΉβπΆ) = (β‘πΉβπ·) β πΆ = π·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f1orn 5487 | . . 3 β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄β1-1-ontoβran πΉ) | |
2 | f1ocnv 5489 | . . 3 β’ (πΉ:π΄β1-1-ontoβran πΉ β β‘πΉ:ran πΉβ1-1-ontoβπ΄) | |
3 | f1of1 5475 | . . 3 β’ (β‘πΉ:ran πΉβ1-1-ontoβπ΄ β β‘πΉ:ran πΉβ1-1βπ΄) | |
4 | f1veqaeq 5786 | . . . 4 β’ ((β‘πΉ:ran πΉβ1-1βπ΄ β§ (πΆ β ran πΉ β§ π· β ran πΉ)) β ((β‘πΉβπΆ) = (β‘πΉβπ·) β πΆ = π·)) | |
5 | 4 | ex 115 | . . 3 β’ (β‘πΉ:ran πΉβ1-1βπ΄ β ((πΆ β ran πΉ β§ π· β ran πΉ) β ((β‘πΉβπΆ) = (β‘πΉβπ·) β πΆ = π·))) |
6 | 1, 2, 3, 5 | 4syl 18 | . 2 β’ (πΉ:π΄β1-1βπ΅ β ((πΆ β ran πΉ β§ π· β ran πΉ) β ((β‘πΉβπΆ) = (β‘πΉβπ·) β πΆ = π·))) |
7 | 6 | imp 124 | 1 β’ ((πΉ:π΄β1-1βπ΅ β§ (πΆ β ran πΉ β§ π· β ran πΉ)) β ((β‘πΉβπΆ) = (β‘πΉβπ·) β πΆ = π·)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1364 β wcel 2160 β‘ccnv 4640 ran crn 4642 β1-1βwf1 5228 β1-1-ontoβwf1o 5230 βcfv 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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