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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 17687 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(〈𝑧, 𝑋〉 · 𝑌)𝑔))))) |
9 | simpl 482 | . . 3 ⊢ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(〈𝑧, 𝑋〉 · 𝑌)𝑔))) → 𝑓 ∈ (𝑋𝐻𝑌)) | |
10 | 8, 9 | syl6bi 253 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) → 𝑓 ∈ (𝑋𝐻𝑌))) |
11 | 10 | ssrdv 3988 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⊆ wss 3948 〈cop 4634 ↦ cmpt 5231 ◡ccnv 5675 Fun wfun 6537 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 Hom chom 17215 compcco 17216 Catccat 17615 Monocmon 17682 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-mon 17684 |
This theorem is referenced by: setcmon 18047 |
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