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Mirrors > Home > MPE Home > Th. List > invffval | Structured version Visualization version GIF version |
Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
invfval.b | β’ π΅ = (BaseβπΆ) |
invfval.n | β’ π = (InvβπΆ) |
invfval.c | β’ (π β πΆ β Cat) |
invfval.x | β’ (π β π β π΅) |
invfval.y | β’ (π β π β π΅) |
invfval.s | β’ π = (SectβπΆ) |
Ref | Expression |
---|---|
invffval | β’ (π β π = (π₯ β π΅, π¦ β π΅ β¦ ((π₯ππ¦) β© β‘(π¦ππ₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.n | . 2 β’ π = (InvβπΆ) | |
2 | invfval.c | . . 3 β’ (π β πΆ β Cat) | |
3 | fveq2 6839 | . . . . . 6 β’ (π = πΆ β (Baseβπ) = (BaseβπΆ)) | |
4 | invfval.b | . . . . . 6 β’ π΅ = (BaseβπΆ) | |
5 | 3, 4 | eqtr4di 2795 | . . . . 5 β’ (π = πΆ β (Baseβπ) = π΅) |
6 | fveq2 6839 | . . . . . . . 8 β’ (π = πΆ β (Sectβπ) = (SectβπΆ)) | |
7 | invfval.s | . . . . . . . 8 β’ π = (SectβπΆ) | |
8 | 6, 7 | eqtr4di 2795 | . . . . . . 7 β’ (π = πΆ β (Sectβπ) = π) |
9 | 8 | oveqd 7368 | . . . . . 6 β’ (π = πΆ β (π₯(Sectβπ)π¦) = (π₯ππ¦)) |
10 | 8 | oveqd 7368 | . . . . . . 7 β’ (π = πΆ β (π¦(Sectβπ)π₯) = (π¦ππ₯)) |
11 | 10 | cnveqd 5829 | . . . . . 6 β’ (π = πΆ β β‘(π¦(Sectβπ)π₯) = β‘(π¦ππ₯)) |
12 | 9, 11 | ineq12d 4171 | . . . . 5 β’ (π = πΆ β ((π₯(Sectβπ)π¦) β© β‘(π¦(Sectβπ)π₯)) = ((π₯ππ¦) β© β‘(π¦ππ₯))) |
13 | 5, 5, 12 | mpoeq123dv 7426 | . . . 4 β’ (π = πΆ β (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ ((π₯(Sectβπ)π¦) β© β‘(π¦(Sectβπ)π₯))) = (π₯ β π΅, π¦ β π΅ β¦ ((π₯ππ¦) β© β‘(π¦ππ₯)))) |
14 | df-inv 17591 | . . . 4 β’ Inv = (π β Cat β¦ (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ ((π₯(Sectβπ)π¦) β© β‘(π¦(Sectβπ)π₯)))) | |
15 | 4 | fvexi 6853 | . . . . 5 β’ π΅ β V |
16 | 15, 15 | mpoex 8004 | . . . 4 β’ (π₯ β π΅, π¦ β π΅ β¦ ((π₯ππ¦) β© β‘(π¦ππ₯))) β V |
17 | 13, 14, 16 | fvmpt 6945 | . . 3 β’ (πΆ β Cat β (InvβπΆ) = (π₯ β π΅, π¦ β π΅ β¦ ((π₯ππ¦) β© β‘(π¦ππ₯)))) |
18 | 2, 17 | syl 17 | . 2 β’ (π β (InvβπΆ) = (π₯ β π΅, π¦ β π΅ β¦ ((π₯ππ¦) β© β‘(π¦ππ₯)))) |
19 | 1, 18 | eqtrid 2789 | 1 β’ (π β π = (π₯ β π΅, π¦ β π΅ β¦ ((π₯ππ¦) β© β‘(π¦ππ₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β© cin 3907 β‘ccnv 5630 βcfv 6493 (class class class)co 7351 β cmpo 7353 Basecbs 17043 Catccat 17504 Sectcsect 17587 Invcinv 17588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-inv 17591 |
This theorem is referenced by: invfval 17602 isoval 17608 |
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