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Mirrors > Home > MPE Home > Th. List > f1ocnvfvrneq | Structured version Visualization version 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 6844 | . . 3 β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄β1-1-ontoβran πΉ) | |
2 | f1ocnv 6845 | . . 3 β’ (πΉ:π΄β1-1-ontoβran πΉ β β‘πΉ:ran πΉβ1-1-ontoβπ΄) | |
3 | f1of1 6832 | . . 3 β’ (β‘πΉ:ran πΉβ1-1-ontoβπ΄ β β‘πΉ:ran πΉβ1-1βπ΄) | |
4 | f1veqaeq 7261 | . . . 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 412 | . . 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 19 | . 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 406 | 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 setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β‘ccnv 5671 ran crn 5673 β1-1βwf1 6539 β1-1-ontoβwf1o 6541 βcfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 |
This theorem is referenced by: (None) |
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