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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 16216 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(〈𝑧, 𝑋〉 · 𝑌)𝑔))))) |
9 | simpl 472 | . . 3 ⊢ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(〈𝑧, 𝑋〉 · 𝑌)𝑔))) → 𝑓 ∈ (𝑋𝐻𝑌)) | |
10 | 8, 9 | syl6bi 242 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) → 𝑓 ∈ (𝑋𝐻𝑌))) |
11 | 10 | ssrdv 3574 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 〈cop 4131 ↦ cmpt 4643 ◡ccnv 5037 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Monocmon 16211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-mon 16213 |
This theorem is referenced by: setcmon 16560 |
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