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Mirrors > Home > MPE Home > Th. List > subcid | Structured version Visualization version GIF version |
Description: The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
subccat.1 | ⊢ 𝐷 = (𝐶 ↾cat 𝐽) |
subccat.j | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subccatid.1 | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
subccatid.2 | ⊢ 1 = (Id‘𝐶) |
subcid.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Ref | Expression |
---|---|
subcid | ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘𝐷)‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subccat.1 | . . . . 5 ⊢ 𝐷 = (𝐶 ↾cat 𝐽) | |
2 | subccat.j | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
3 | subccatid.1 | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
4 | subccatid.2 | . . . . 5 ⊢ 1 = (Id‘𝐶) | |
5 | 1, 2, 3, 4 | subccatid 17104 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑥 ∈ 𝑆 ↦ ( 1 ‘𝑥)))) |
6 | 5 | simprd 496 | . . 3 ⊢ (𝜑 → (Id‘𝐷) = (𝑥 ∈ 𝑆 ↦ ( 1 ‘𝑥))) |
7 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
8 | 7 | fveq2d 6667 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋)) |
9 | subcid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
10 | fvexd 6678 | . . 3 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ V) | |
11 | 6, 8, 9, 10 | fvmptd 6767 | . 2 ⊢ (𝜑 → ((Id‘𝐷)‘𝑋) = ( 1 ‘𝑋)) |
12 | 11 | eqcomd 2824 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘𝐷)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 × cxp 5546 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 Catccat 16923 Idccid 16924 ↾cat cresc 17066 Subcatcsubc 17067 |
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-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-er 8278 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 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-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-hom 16577 df-cco 16578 df-cat 16927 df-cid 16928 df-homf 16929 df-ssc 17068 df-resc 17069 df-subc 17070 |
This theorem is referenced by: subsubc 17111 funcres 17154 funcres2b 17155 rngcid 44178 ringcid 44224 |
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