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Mirrors > Home > MPE Home > Th. List > comfeqd | Structured version Visualization version GIF version |
Description: Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfeqd.1 | ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷)) |
comfeqd.2 | ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
Ref | Expression |
---|---|
comfeqd | ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfeqd.1 | . . . . . . . . 9 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷)) | |
2 | 1 | oveqd 7167 | . . . . . . . 8 ⊢ (𝜑 → (〈𝑥, 𝑦〉(comp‘𝐶)𝑧) = (〈𝑥, 𝑦〉(comp‘𝐷)𝑧)) |
3 | 2 | oveqd 7167 | . . . . . . 7 ⊢ (𝜑 → (𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
4 | 3 | ralrimivw 3183 | . . . . . 6 ⊢ (𝜑 → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
5 | 4 | ralrimivw 3183 | . . . . 5 ⊢ (𝜑 → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
6 | 5 | ralrimivw 3183 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
7 | 6 | ralrimivw 3183 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
8 | 7 | ralrimivw 3183 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
9 | eqid 2821 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
10 | eqid 2821 | . . 3 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
11 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqidd 2822 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶)) | |
13 | comfeqd.2 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
14 | 13 | homfeqbas 16960 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
15 | 9, 10, 11, 12, 14, 13 | comfeq 16970 | . 2 ⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓))) |
16 | 8, 15 | mpbird 259 | 1 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∀wral 3138 〈cop 4566 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 compcco 16571 Homf chomf 16931 compfccomf 16932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-homf 16935 df-comf 16936 |
This theorem is referenced by: fullresc 17115 |
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