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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 7148 | . 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 17426 | . . . . 5 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐼‘𝑥), (𝐽‘𝑦)〉))) |
11 | 10 | simprd 496 | . . . 4 ⊢ (𝜑 → (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐼‘𝑥), (𝐽‘𝑦)〉)) |
12 | 2, 11 | syl5eq 2865 | . . 3 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐼‘𝑥), (𝐽‘𝑦)〉)) |
13 | simprl 767 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
14 | 13 | fveq2d 6667 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐼‘𝑥) = (𝐼‘𝑅)) |
15 | simprr 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
16 | 15 | fveq2d 6667 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐽‘𝑦) = (𝐽‘𝑆)) |
17 | 14, 16 | opeq12d 4803 | . . 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 5347 | . . . 4 ⊢ 〈(𝐼‘𝑅), (𝐽‘𝑆)〉 ∈ V | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → 〈(𝐼‘𝑅), (𝐽‘𝑆)〉 ∈ V) |
22 | 12, 17, 18, 19, 21 | ovmpod 7291 | . 2 ⊢ (𝜑 → (𝑅 1 𝑆) = 〈(𝐼‘𝑅), (𝐽‘𝑆)〉) |
23 | 1, 22 | syl5eqr 2867 | 1 ⊢ (𝜑 → ( 1 ‘〈𝑅, 𝑆〉) = 〈(𝐼‘𝑅), (𝐽‘𝑆)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 Basecbs 16471 Catccat 16923 Idccid 16924 ×c cxpc 17406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-hom 16577 df-cco 16578 df-cat 16927 df-cid 16928 df-xpc 17410 |
This theorem is referenced by: 1stfcl 17435 2ndfcl 17436 prfcl 17441 evlfcl 17460 curf1cl 17466 curfcl 17470 hofcl 17497 |
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