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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 7359 | . 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 18075 | . . . . 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 | eqtrid 2788 | . . 3 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐼‘𝑥), (𝐽‘𝑦)〉)) |
13 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
14 | 13 | fveq2d 6846 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐼‘𝑥) = (𝐼‘𝑅)) |
15 | simprr 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
16 | 15 | fveq2d 6846 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐽‘𝑦) = (𝐽‘𝑆)) |
17 | 14, 16 | opeq12d 4838 | . . 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 5421 | . . . 4 ⊢ 〈(𝐼‘𝑅), (𝐽‘𝑆)〉 ∈ V | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → 〈(𝐼‘𝑅), (𝐽‘𝑆)〉 ∈ V) |
22 | 12, 17, 18, 19, 21 | ovmpod 7506 | . 2 ⊢ (𝜑 → (𝑅 1 𝑆) = 〈(𝐼‘𝑅), (𝐽‘𝑆)〉) |
23 | 1, 22 | eqtr3id 2790 | 1 ⊢ (𝜑 → ( 1 ‘〈𝑅, 𝑆〉) = 〈(𝐼‘𝑅), (𝐽‘𝑆)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 〈cop 4592 ‘cfv 6496 (class class class)co 7356 ∈ cmpo 7358 Basecbs 17082 Catccat 17543 Idccid 17544 ×c cxpc 18055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-struct 17018 df-slot 17053 df-ndx 17065 df-base 17083 df-hom 17156 df-cco 17157 df-cat 17547 df-cid 17548 df-xpc 18059 |
This theorem is referenced by: 1stfcl 18084 2ndfcl 18085 prfcl 18090 evlfcl 18110 curf1cl 18116 curfcl 18120 hofcl 18147 |
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