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Mirrors > Home > MPE Home > Th. List > homarel | Structured version Visualization version GIF version |
Description: An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | β’ π» = (HomaβπΆ) |
Ref | Expression |
---|---|
homarel | β’ Rel (ππ»π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 5683 | . . . 4 β’ (((BaseβπΆ) Γ (BaseβπΆ)) Γ V) β (V Γ V) | |
2 | homahom.h | . . . . . . 7 β’ π» = (HomaβπΆ) | |
3 | eqid 2724 | . . . . . . 7 β’ (BaseβπΆ) = (BaseβπΆ) | |
4 | 2 | homarcl 17982 | . . . . . . 7 β’ (π β (ππ»π) β πΆ β Cat) |
5 | 2, 3, 4 | homaf 17984 | . . . . . 6 β’ (π β (ππ»π) β π»:((BaseβπΆ) Γ (BaseβπΆ))βΆπ« (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) |
6 | 2, 3 | homarcl2 17989 | . . . . . . 7 β’ (π β (ππ»π) β (π β (BaseβπΆ) β§ π β (BaseβπΆ))) |
7 | 6 | simpld 494 | . . . . . 6 β’ (π β (ππ»π) β π β (BaseβπΆ)) |
8 | 6 | simprd 495 | . . . . . 6 β’ (π β (ππ»π) β π β (BaseβπΆ)) |
9 | 5, 7, 8 | fovcdmd 7573 | . . . . 5 β’ (π β (ππ»π) β (ππ»π) β π« (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) |
10 | elelpwi 4605 | . . . . 5 β’ ((π β (ππ»π) β§ (ππ»π) β π« (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) β π β (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) | |
11 | 9, 10 | mpdan 684 | . . . 4 β’ (π β (ππ»π) β π β (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) |
12 | 1, 11 | sselid 3973 | . . 3 β’ (π β (ππ»π) β π β (V Γ V)) |
13 | 12 | ssriv 3979 | . 2 β’ (ππ»π) β (V Γ V) |
14 | df-rel 5674 | . 2 β’ (Rel (ππ»π) β (ππ»π) β (V Γ V)) | |
15 | 13, 14 | mpbir 230 | 1 β’ Rel (ππ»π) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 Vcvv 3466 β wss 3941 π« cpw 4595 Γ cxp 5665 Rel wrel 5672 βcfv 6534 (class class class)co 7402 Basecbs 17145 Homachoma 17977 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-homa 17980 |
This theorem is referenced by: homahom 17993 homadm 17994 homacd 17995 homadmcd 17996 |
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