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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 5484 | . . 3 β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄β1-1-ontoβran πΉ) | |
2 | f1ocnv 5486 | . . 3 β’ (πΉ:π΄β1-1-ontoβran πΉ β β‘πΉ:ran πΉβ1-1-ontoβπ΄) | |
3 | f1of1 5472 | . . 3 β’ (β‘πΉ:ran πΉβ1-1-ontoβπ΄ β β‘πΉ:ran πΉβ1-1βπ΄) | |
4 | f1veqaeq 5783 | . . . 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 1363 β wcel 2158 β‘ccnv 4637 ran crn 4639 β1-1βwf1 5225 β1-1-ontoβwf1o 5227 βcfv 5228 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 |
This theorem is referenced by: (None) |
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