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Mirrors > Home > MPE Home > Th. List > monhom | Structured version Visualization version GIF version |
Description: A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
ismon.b | ⊢ 𝐵 = (Base‘𝐶) |
ismon.h | ⊢ 𝐻 = (Hom ‘𝐶) |
ismon.o | ⊢ · = (comp‘𝐶) |
ismon.s | ⊢ 𝑀 = (Mono‘𝐶) |
ismon.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
ismon.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ismon.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
monhom | ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismon.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | ismon.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | ismon.o | . . . 4 ⊢ · = (comp‘𝐶) | |
4 | ismon.s | . . . 4 ⊢ 𝑀 = (Mono‘𝐶) | |
5 | ismon.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
6 | ismon.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | ismon.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismon 16873 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(〈𝑧, 𝑋〉 · 𝑌)𝑔))))) |
9 | simpl 475 | . . 3 ⊢ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(〈𝑧, 𝑋〉 · 𝑌)𝑔))) → 𝑓 ∈ (𝑋𝐻𝑌)) | |
10 | 8, 9 | syl6bi 245 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) → 𝑓 ∈ (𝑋𝐻𝑌))) |
11 | 10 | ssrdv 3866 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3090 ⊆ wss 3831 〈cop 4450 ↦ cmpt 5013 ◡ccnv 5410 Fun wfun 6187 ‘cfv 6193 (class class class)co 6982 Basecbs 16345 Hom chom 16438 compcco 16439 Catccat 16805 Monocmon 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-ov 6985 df-oprab 6986 df-mpo 6987 df-1st 7507 df-2nd 7508 df-mon 16870 |
This theorem is referenced by: setcmon 17217 |
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