![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7361 | . 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 18081 | . . . . 5 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩))) |
11 | 10 | simprd 497 | . . . 4 ⊢ (𝜑 → (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)) |
12 | 2, 11 | eqtrid 2785 | . . 3 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)) |
13 | simprl 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
14 | 13 | fveq2d 6847 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐼‘𝑥) = (𝐼‘𝑅)) |
15 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
16 | 15 | fveq2d 6847 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝐽‘𝑦) = (𝐽‘𝑆)) |
17 | 14, 16 | opeq12d 4839 | . . 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 5422 | . . . 4 ⊢ ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩ ∈ V | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩ ∈ V) |
22 | 12, 17, 18, 19, 21 | ovmpod 7508 | . 2 ⊢ (𝜑 → (𝑅 1 𝑆) = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩) |
23 | 1, 22 | eqtr3id 2787 | 1 ⊢ (𝜑 → ( 1 ‘⟨𝑅, 𝑆⟩) = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⟨cop 4593 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 Basecbs 17088 Catccat 17549 Idccid 17550 ×c cxpc 18061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-hom 17162 df-cco 17163 df-cat 17553 df-cid 17554 df-xpc 18065 |
This theorem is referenced by: 1stfcl 18090 2ndfcl 18091 prfcl 18096 evlfcl 18116 curf1cl 18122 curfcl 18126 hofcl 18153 |
Copyright terms: Public domain | W3C validator |