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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 6796 | . . 3 β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄β1-1-ontoβran πΉ) | |
2 | f1ocnv 6797 | . . 3 β’ (πΉ:π΄β1-1-ontoβran πΉ β β‘πΉ:ran πΉβ1-1-ontoβπ΄) | |
3 | f1of1 6784 | . . 3 β’ (β‘πΉ:ran πΉβ1-1-ontoβπ΄ β β‘πΉ:ran πΉβ1-1βπ΄) | |
4 | f1veqaeq 7205 | . . . 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 414 | . . 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 408 | 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 397 = wceq 1542 β wcel 2107 β‘ccnv 5633 ran crn 5635 β1-1βwf1 6494 β1-1-ontoβwf1o 6496 βcfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 |
This theorem is referenced by: (None) |
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