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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 | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homarel | ⊢ Rel (𝑋𝐻𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 5689 | . . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V) | |
2 | homahom.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 2 | homarcl 18043 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
5 | 2, 3, 4 | homaf 18045 | . . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
6 | 2, 3 | homarcl2 18050 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
7 | 6 | simpld 493 | . . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶)) |
8 | 6 | simprd 494 | . . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶)) |
9 | 5, 7, 8 | fovcdmd 7588 | . . . . 5 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
10 | elelpwi 4608 | . . . . 5 ⊢ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V)) | |
11 | 9, 10 | mpdan 685 | . . . 4 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V)) |
12 | 1, 11 | sselid 3977 | . . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V)) |
13 | 12 | ssriv 3983 | . 2 ⊢ (𝑋𝐻𝑌) ⊆ (V × V) |
14 | df-rel 5680 | . 2 ⊢ (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V)) | |
15 | 13, 14 | mpbir 230 | 1 ⊢ Rel (𝑋𝐻𝑌) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3463 ⊆ wss 3947 𝒫 cpw 4598 × cxp 5671 Rel wrel 5678 ‘cfv 6544 (class class class)co 7414 Basecbs 17206 Homachoma 18038 |
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 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-homa 18041 |
This theorem is referenced by: homahom 18054 homadm 18055 homacd 18056 homadmcd 18057 |
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