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Mirrors > Home > MPE Home > Th. List > rescbas | Structured version Visualization version GIF version |
Description: Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rescbas.d | ⊢ 𝐷 = (𝐶 ↾cat 𝐻) |
rescbas.b | ⊢ 𝐵 = (Base‘𝐶) |
rescbas.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
rescbas.h | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
rescbas.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Ref | Expression |
---|---|
rescbas | ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 16535 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
2 | 1re 10630 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1nn 11636 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | 4nn0 11904 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
5 | 1nn0 11901 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12225 | . . . . . 6 ⊢ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 12124 | . . . . 5 ⊢ 1 < ;14 |
8 | 2, 7 | ltneii 10742 | . . . 4 ⊢ 1 ≠ ;14 |
9 | basendx 16539 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
10 | homndx 16679 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
11 | 9, 10 | neeq12i 3053 | . . . 4 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
12 | 8, 11 | mpbir 234 | . . 3 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
13 | 1, 12 | setsnid 16531 | . 2 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
14 | rescbas.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
15 | eqid 2798 | . . . 4 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆) | |
16 | rescbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
17 | 15, 16 | ressbas2 16547 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘(𝐶 ↾s 𝑆))) |
18 | 14, 17 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 = (Base‘(𝐶 ↾s 𝑆))) |
19 | rescbas.d | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
20 | rescbas.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
21 | 16 | fvexi 6659 | . . . . . 6 ⊢ 𝐵 ∈ V |
22 | 21 | ssex 5189 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ V) |
23 | 14, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
24 | rescbas.h | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
25 | 19, 20, 23, 24 | rescval2 17090 | . . 3 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
26 | 25 | fveq2d 6649 | . 2 ⊢ (𝜑 → (Base‘𝐷) = (Base‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
27 | 13, 18, 26 | 3eqtr4a 2859 | 1 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⊆ wss 3881 〈cop 4531 × cxp 5517 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 1c1 10527 4c4 11682 ;cdc 12086 ndxcnx 16472 sSet csts 16473 Basecbs 16475 ↾s cress 16476 Hom chom 16568 ↾cat cresc 17070 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-hom 16581 df-resc 17073 |
This theorem is referenced by: reschomf 17093 subccatid 17108 issubc3 17111 fullresc 17113 funcres 17158 funcres2b 17159 funcres2 17160 rngcbas 44589 ringcbas 44635 |
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