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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 16545 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
2 | 1re 10643 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1nn 11651 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | 4nn0 11919 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
5 | 1nn0 11916 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12240 | . . . . . 6 ⊢ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 12139 | . . . . 5 ⊢ 1 < ;14 |
8 | 2, 7 | ltneii 10755 | . . . 4 ⊢ 1 ≠ ;14 |
9 | basendx 16549 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
10 | homndx 16689 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
11 | 9, 10 | neeq12i 3084 | . . . 4 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
12 | 8, 11 | mpbir 233 | . . 3 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
13 | 1, 12 | setsnid 16541 | . 2 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
14 | rescbas.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
15 | eqid 2823 | . . . 4 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆) | |
16 | rescbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
17 | 15, 16 | ressbas2 16557 | . . 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 6686 | . . . . . 6 ⊢ 𝐵 ∈ V |
22 | 21 | ssex 5227 | . . . . 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 17100 | . . 3 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
26 | 25 | fveq2d 6676 | . 2 ⊢ (𝜑 → (Base‘𝐷) = (Base‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
27 | 13, 18, 26 | 3eqtr4a 2884 | 1 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ⊆ wss 3938 〈cop 4575 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 1c1 10540 4c4 11697 ;cdc 12101 ndxcnx 16482 sSet csts 16483 Basecbs 16485 ↾s cress 16486 Hom chom 16578 ↾cat cresc 17080 |
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-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-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-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 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-resc 17083 |
This theorem is referenced by: reschomf 17103 subccatid 17118 issubc3 17121 fullresc 17123 funcres 17168 funcres2b 17169 funcres2 17170 rngcbas 44243 ringcbas 44289 |
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