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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 6881 | . . . . . 6 β’ (π = πΆ β (Baseβπ) = (BaseβπΆ)) | |
4 | invfval.b | . . . . . 6 β’ π΅ = (BaseβπΆ) | |
5 | 3, 4 | eqtr4di 2782 | . . . . 5 β’ (π = πΆ β (Baseβπ) = π΅) |
6 | fveq2 6881 | . . . . . . . 8 β’ (π = πΆ β (Sectβπ) = (SectβπΆ)) | |
7 | invfval.s | . . . . . . . 8 β’ π = (SectβπΆ) | |
8 | 6, 7 | eqtr4di 2782 | . . . . . . 7 β’ (π = πΆ β (Sectβπ) = π) |
9 | 8 | oveqd 7418 | . . . . . 6 β’ (π = πΆ β (π₯(Sectβπ)π¦) = (π₯ππ¦)) |
10 | 8 | oveqd 7418 | . . . . . . 7 β’ (π = πΆ β (π¦(Sectβπ)π₯) = (π¦ππ₯)) |
11 | 10 | cnveqd 5865 | . . . . . 6 β’ (π = πΆ β β‘(π¦(Sectβπ)π₯) = β‘(π¦ππ₯)) |
12 | 9, 11 | ineq12d 4205 | . . . . 5 β’ (π = πΆ β ((π₯(Sectβπ)π¦) β© β‘(π¦(Sectβπ)π₯)) = ((π₯ππ¦) β© β‘(π¦ππ₯))) |
13 | 5, 5, 12 | mpoeq123dv 7476 | . . . 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 17691 | . . . 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 6895 | . . . . 5 β’ π΅ β V |
16 | 15, 15 | mpoex 8059 | . . . 4 β’ (π₯ β π΅, π¦ β π΅ β¦ ((π₯ππ¦) β© β‘(π¦ππ₯))) β V |
17 | 13, 14, 16 | fvmpt 6988 | . . 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 2776 | 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 1533 β wcel 2098 β© cin 3939 β‘ccnv 5665 βcfv 6533 (class class class)co 7401 β cmpo 7403 Basecbs 17140 Catccat 17604 Sectcsect 17687 Invcinv 17688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-inv 17691 |
This theorem is referenced by: invfval 17702 isoval 17708 |
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