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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 6566 | . . . . . . . 8 ⊢ (𝜑 → (〈𝑥, 𝑦〉(comp‘𝐶)𝑧) = (〈𝑥, 𝑦〉(comp‘𝐷)𝑧)) |
3 | 2 | oveqd 6566 | . . . . . . 7 ⊢ (𝜑 → (𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
4 | 3 | ralrimivw 2950 | . . . . . 6 ⊢ (𝜑 → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑓)) |
5 | 4 | ralrimivw 2950 | . . . . 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 2950 | . . . 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 2950 | . . 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 2950 | . 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 2610 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
10 | eqid 2610 | . . 3 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
11 | eqid 2610 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqidd 2611 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶)) | |
13 | comfeqd.2 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
14 | 13 | homfeqbas 16179 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
15 | 9, 10, 11, 12, 14, 13 | comfeq 16189 | . 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 246 | 1 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∀wral 2896 〈cop 4131 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 Homf chomf 16150 compfccomf 16151 |
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-homf 16154 df-comf 16155 |
This theorem is referenced by: fullresc 16334 |
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