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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 5573 | . . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V) | |
2 | homahom.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | eqid 2823 | . . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 2 | homarcl 17290 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
5 | 2, 3, 4 | homaf 17292 | . . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
6 | 2, 3 | homarcl2 17297 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
7 | 6 | simpld 497 | . . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶)) |
8 | 6 | simprd 498 | . . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶)) |
9 | 5, 7, 8 | fovrnd 7322 | . . . . 5 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
10 | elelpwi 4553 | . . . . 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 | sseldi 3967 | . . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V)) |
13 | 12 | ssriv 3973 | . 2 ⊢ (𝑋𝐻𝑌) ⊆ (V × V) |
14 | df-rel 5564 | . 2 ⊢ (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V)) | |
15 | 13, 14 | mpbir 233 | 1 ⊢ Rel (𝑋𝐻𝑌) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 × cxp 5555 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Homachoma 17285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-homa 17288 |
This theorem is referenced by: homahom 17301 homadm 17302 homacd 17303 homadmcd 17304 |
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