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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 7448 | . . . . . . . 8 ⊢ (𝜑 → (〈𝑥, 𝑦〉(comp‘𝐶)𝑧) = (〈𝑥, 𝑦〉(comp‘𝐷)𝑧)) |
3 | 2 | oveqd 7448 | . . . . . . 7 ⊢ (𝜑 → (𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
4 | 3 | ralrimivw 3148 | . . . . . 6 ⊢ (𝜑 → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
5 | 4 | ralrimivw 3148 | . . . . 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 3148 | . . . 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 3148 | . . 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 3148 | . 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 2735 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
10 | eqid 2735 | . . 3 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
11 | eqid 2735 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqidd 2736 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶)) | |
13 | comfeqd.2 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
14 | 13 | homfeqbas 17741 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
15 | 9, 10, 11, 12, 14, 13 | comfeq 17751 | . 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 257 | 1 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∀wral 3059 〈cop 4637 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 compcco 17310 Homf chomf 17711 compfccomf 17712 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-homf 17715 df-comf 17716 |
This theorem is referenced by: fullresc 17902 |
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