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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 17729 | . . . 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 6844 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋)) |
9 | subcid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
10 | fvexd 6855 | . . 3 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ V) | |
11 | 6, 8, 9, 10 | fvmptd 6953 | . 2 ⊢ (𝜑 → ((Id‘𝐷)‘𝑋) = ( 1 ‘𝑋)) |
12 | 11 | eqcomd 2742 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘𝐷)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3444 ↦ cmpt 5187 × cxp 5630 Fn wfn 6489 ‘cfv 6494 (class class class)co 7354 Catccat 17541 Idccid 17542 ↾cat cresc 17688 Subcatcsubc 17689 |
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 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
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 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-pm 8765 df-ixp 8833 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-hom 17154 df-cco 17155 df-cat 17545 df-cid 17546 df-homf 17547 df-ssc 17690 df-resc 17691 df-subc 17692 |
This theorem is referenced by: subsubc 17736 funcres 17779 funcres2b 17780 rngcid 46247 ringcid 46293 |
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