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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 17118 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑥 ∈ 𝑆 ↦ ( 1 ‘𝑥)))) |
6 | 5 | simprd 498 | . . 3 ⊢ (𝜑 → (Id‘𝐷) = (𝑥 ∈ 𝑆 ↦ ( 1 ‘𝑥))) |
7 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
8 | 7 | fveq2d 6676 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋)) |
9 | subcid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
10 | fvexd 6687 | . . 3 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ V) | |
11 | 6, 8, 9, 10 | fvmptd 6777 | . 2 ⊢ (𝜑 → ((Id‘𝐷)‘𝑋) = ( 1 ‘𝑋)) |
12 | 11 | eqcomd 2829 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘𝐷)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 Catccat 16937 Idccid 16938 ↾cat cresc 17080 Subcatcsubc 17081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-hom 16591 df-cco 16592 df-cat 16941 df-cid 16942 df-homf 16943 df-ssc 17082 df-resc 17083 df-subc 17084 |
This theorem is referenced by: subsubc 17125 funcres 17168 funcres2b 17169 rngcid 44257 ringcid 44303 |
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