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Mirrors > Home > MPE Home > Th. List > Mathboxes > rfovcnvfvd | Structured version Visualization version GIF version |
Description: Value of the converse of the operator, (π΄ππ΅), which maps between relations and functions for relations between base sets, π΄ and π΅, evaluated at function πΊ. (Contributed by RP, 27-Apr-2021.) |
Ref | Expression |
---|---|
rfovd.rf | β’ π = (π β V, π β V β¦ (π β π« (π Γ π) β¦ (π₯ β π β¦ {π¦ β π β£ π₯ππ¦}))) |
rfovd.a | β’ (π β π΄ β π) |
rfovd.b | β’ (π β π΅ β π) |
rfovcnvf1od.f | β’ πΉ = (π΄ππ΅) |
rfovcnvfv.g | β’ (π β πΊ β (π« π΅ βm π΄)) |
Ref | Expression |
---|---|
rfovcnvfvd | β’ (π β (β‘πΉβπΊ) = {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πΊβπ₯))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rfovd.rf | . . 3 β’ π = (π β V, π β V β¦ (π β π« (π Γ π) β¦ (π₯ β π β¦ {π¦ β π β£ π₯ππ¦}))) | |
2 | rfovd.a | . . 3 β’ (π β π΄ β π) | |
3 | rfovd.b | . . 3 β’ (π β π΅ β π) | |
4 | rfovcnvf1od.f | . . 3 β’ πΉ = (π΄ππ΅) | |
5 | 1, 2, 3, 4 | rfovcnvd 43245 | . 2 β’ (π β β‘πΉ = (π β (π« π΅ βm π΄) β¦ {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πβπ₯))})) |
6 | fveq1 6880 | . . . . . 6 β’ (π = πΊ β (πβπ₯) = (πΊβπ₯)) | |
7 | 6 | eleq2d 2811 | . . . . 5 β’ (π = πΊ β (π¦ β (πβπ₯) β π¦ β (πΊβπ₯))) |
8 | 7 | anbi2d 628 | . . . 4 β’ (π = πΊ β ((π₯ β π΄ β§ π¦ β (πβπ₯)) β (π₯ β π΄ β§ π¦ β (πΊβπ₯)))) |
9 | 8 | opabbidv 5204 | . . 3 β’ (π = πΊ β {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πβπ₯))} = {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πΊβπ₯))}) |
10 | 9 | adantl 481 | . 2 β’ ((π β§ π = πΊ) β {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πβπ₯))} = {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πΊβπ₯))}) |
11 | rfovcnvfv.g | . 2 β’ (π β πΊ β (π« π΅ βm π΄)) | |
12 | simprl 768 | . . 3 β’ ((π β§ (π₯ β π΄ β§ π¦ β (πΊβπ₯))) β π₯ β π΄) | |
13 | elmapi 8839 | . . . . . . . 8 β’ (πΊ β (π« π΅ βm π΄) β πΊ:π΄βΆπ« π΅) | |
14 | 13 | ffvelcdmda 7076 | . . . . . . 7 β’ ((πΊ β (π« π΅ βm π΄) β§ π₯ β π΄) β (πΊβπ₯) β π« π΅) |
15 | 11, 14 | sylan 579 | . . . . . 6 β’ ((π β§ π₯ β π΄) β (πΊβπ₯) β π« π΅) |
16 | 15 | elpwid 4603 | . . . . 5 β’ ((π β§ π₯ β π΄) β (πΊβπ₯) β π΅) |
17 | 16 | sseld 3973 | . . . 4 β’ ((π β§ π₯ β π΄) β (π¦ β (πΊβπ₯) β π¦ β π΅)) |
18 | 17 | impr 454 | . . 3 β’ ((π β§ (π₯ β π΄ β§ π¦ β (πΊβπ₯))) β π¦ β π΅) |
19 | 2, 3, 12, 18 | opabex2 8036 | . 2 β’ (π β {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πΊβπ₯))} β V) |
20 | 5, 10, 11, 19 | fvmptd 6995 | 1 β’ (π β (β‘πΉβπΊ) = {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πΊβπ₯))}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3424 Vcvv 3466 π« cpw 4594 class class class wbr 5138 {copab 5200 β¦ cmpt 5221 Γ cxp 5664 β‘ccnv 5665 βcfv 6533 (class class class)co 7401 β cmpo 7403 βm cmap 8816 |
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-map 8818 |
This theorem is referenced by: (None) |
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