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Mirrors > Home > MPE Home > Th. List > xpcid | Structured version Visualization version GIF version |
Description: The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
xpccat.t | ⊢ 𝑇 = (𝐶 ×c 𝐷) |
xpccat.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
xpccat.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
xpccat.x | ⊢ 𝑋 = (Base‘𝐶) |
xpccat.y | ⊢ 𝑌 = (Base‘𝐷) |
xpccat.i | ⊢ 𝐼 = (Id‘𝐶) |
xpccat.j | ⊢ 𝐽 = (Id‘𝐷) |
xpcid.1 | ⊢ 1 = (Id‘𝑇) |
xpcid.r | ⊢ (𝜑 → 𝑅 ∈ 𝑋) |
xpcid.s | ⊢ (𝜑 → 𝑆 ∈ 𝑌) |
Ref | Expression |
---|---|
xpcid | ⊢ (𝜑 → ( 1 ‘⟨𝑅, 𝑆⟩) = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7405 | . 2 ⊢ (𝑅 1 𝑆) = ( 1 ‘⟨𝑅, 𝑆⟩) | |
2 | xpcid.1 | . . . 4 ⊢ 1 = (Id‘𝑇) | |
3 | xpccat.t | . . . . . 6 ⊢ 𝑇 = (𝐶 ×c 𝐷) | |
4 | xpccat.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | xpccat.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
6 | xpccat.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐶) | |
7 | xpccat.y | . . . . . 6 ⊢ 𝑌 = (Base‘𝐷) | |
8 | xpccat.i | . . . . . 6 ⊢ 𝐼 = (Id‘𝐶) | |
9 | xpccat.j | . . . . . 6 ⊢ 𝐽 = (Id‘𝐷) | |
10 | 3, 4, 5, 6, 7, 8, 9 | xpccatid 18148 | . . . . 5 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩))) |
11 | 10 | simprd 495 | . . . 4 ⊢ (𝜑 → (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)) |
12 | 2, 11 | eqtrid 2776 | . . 3 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)) |
13 | simprl 768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
14 | 13 | fveq2d 6886 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐼‘𝑥) = (𝐼‘𝑅)) |
15 | simprr 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
16 | 15 | fveq2d 6886 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐽‘𝑦) = (𝐽‘𝑆)) |
17 | 14, 16 | opeq12d 4874 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩ = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩) |
18 | xpcid.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑋) | |
19 | xpcid.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑌) | |
20 | opex 5455 | . . . 4 ⊢ ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩ ∈ V | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩ ∈ V) |
22 | 12, 17, 18, 19, 21 | ovmpod 7553 | . 2 ⊢ (𝜑 → (𝑅 1 𝑆) = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩) |
23 | 1, 22 | eqtr3id 2778 | 1 ⊢ (𝜑 → ( 1 ‘⟨𝑅, 𝑆⟩) = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⟨cop 4627 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 Basecbs 17149 Catccat 17613 Idccid 17614 ×c cxpc 18128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-hom 17226 df-cco 17227 df-cat 17617 df-cid 17618 df-xpc 18132 |
This theorem is referenced by: 1stfcl 18157 2ndfcl 18158 prfcl 18163 evlfcl 18183 curf1cl 18189 curfcl 18193 hofcl 18220 |
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