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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 5688 | . . . 4 β’ (((BaseβπΆ) Γ (BaseβπΆ)) Γ V) β (V Γ V) | |
2 | homahom.h | . . . . . . 7 β’ π» = (HomaβπΆ) | |
3 | eqid 2728 | . . . . . . 7 β’ (BaseβπΆ) = (BaseβπΆ) | |
4 | 2 | homarcl 18010 | . . . . . . 7 β’ (π β (ππ»π) β πΆ β Cat) |
5 | 2, 3, 4 | homaf 18012 | . . . . . 6 β’ (π β (ππ»π) β π»:((BaseβπΆ) Γ (BaseβπΆ))βΆπ« (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) |
6 | 2, 3 | homarcl2 18017 | . . . . . . 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 7587 | . . . . 5 β’ (π β (ππ»π) β (ππ»π) β π« (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) |
10 | elelpwi 4608 | . . . . 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 686 | . . . 4 β’ (π β (ππ»π) β π β (((BaseβπΆ) Γ (BaseβπΆ)) Γ V)) |
12 | 1, 11 | sselid 3976 | . . 3 β’ (π β (ππ»π) β π β (V Γ V)) |
13 | 12 | ssriv 3982 | . 2 β’ (ππ»π) β (V Γ V) |
14 | df-rel 5679 | . 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 1534 β wcel 2099 Vcvv 3470 β wss 3945 π« cpw 4598 Γ cxp 5670 Rel wrel 5677 βcfv 6542 (class class class)co 7414 Basecbs 17173 Homachoma 18005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-homa 18008 |
This theorem is referenced by: homahom 18021 homadm 18022 homacd 18023 homadmcd 18024 |
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