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Mirrors > Home > MPE Home > Th. List > subcss2 | Structured version Visualization version GIF version |
Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
subcss1.1 | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subcss1.2 | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
subcss2.h | ⊢ 𝐻 = (Hom ‘𝐶) |
subcss2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
subcss2.y | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
Ref | Expression |
---|---|
subcss2 | ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcss1.2 | . . 3 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
2 | subcss1.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
3 | eqid 2724 | . . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
4 | 2, 3 | subcssc 17789 | . . 3 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶)) |
5 | subcss2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
6 | subcss2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
7 | 1, 4, 5, 6 | ssc2 17768 | . 2 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf ‘𝐶)𝑌)) |
8 | eqid 2724 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | subcss2.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
10 | 2, 1, 8 | subcss1 17791 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶)) |
11 | 10, 5 | sseldd 3975 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
12 | 10, 6 | sseldd 3975 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
13 | 3, 8, 9, 11, 12 | homfval 17635 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌)) |
14 | 7, 13 | sseqtrd 4014 | 1 ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 × cxp 5664 Fn wfn 6528 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 Hom chom 17207 Homf chomf 17609 Subcatcsubc 17755 |
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-pm 8819 df-ixp 8888 df-homf 17613 df-ssc 17756 df-subc 17758 |
This theorem is referenced by: subccatid 17795 funcres 17845 funcres2b 17846 subthinc 47848 |
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