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Mirrors > Home > MPE Home > Th. List > resslem | Structured version Visualization version GIF version |
Description: Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
resslem.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
resslem.e | ⊢ 𝐶 = (𝐸‘𝑊) |
resslem.f | ⊢ 𝐸 = Slot 𝑁 |
resslem.n | ⊢ 𝑁 ∈ ℕ |
resslem.b | ⊢ 1 < 𝑁 |
Ref | Expression |
---|---|
resslem | ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resslem.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
2 | eqid 2651 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | 1, 2 | ressid2 15975 | . . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
4 | 3 | fveq2d 6233 | . . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
5 | 4 | 3expib 1287 | . . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
6 | 1, 2 | ressval2 15976 | . . . . . . 7 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
7 | 6 | fveq2d 6233 | . . . . . 6 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
8 | resslem.f | . . . . . . . 8 ⊢ 𝐸 = Slot 𝑁 | |
9 | resslem.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
10 | 8, 9 | ndxid 15930 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
11 | 8, 9 | ndxarg 15929 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
12 | 1re 10077 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
13 | resslem.b | . . . . . . . . . 10 ⊢ 1 < 𝑁 | |
14 | 12, 13 | gtneii 10187 | . . . . . . . . 9 ⊢ 𝑁 ≠ 1 |
15 | 11, 14 | eqnetri 2893 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 1 |
16 | basendx 15970 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
17 | 15, 16 | neeqtrri 2896 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Base‘ndx) |
18 | 10, 17 | setsnid 15962 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
19 | 7, 18 | syl6eqr 2703 | . . . . 5 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
20 | 19 | 3expib 1287 | . . . 4 ⊢ (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
21 | 5, 20 | pm2.61i 176 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
22 | reldmress 15973 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
23 | 22 | ovprc1 6724 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
24 | 1, 23 | syl5eq 2697 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
25 | 24 | fveq2d 6233 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
26 | 8 | str0 15958 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
27 | 25, 26 | syl6eqr 2703 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
28 | fvprc 6223 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
29 | 27, 28 | eqtr4d 2688 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
30 | 29 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
31 | 21, 30 | pm2.61ian 848 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
32 | resslem.e | . 2 ⊢ 𝐶 = (𝐸‘𝑊) | |
33 | 31, 32 | syl6reqr 2704 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 〈cop 4216 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 1c1 9975 < clt 10112 ℕcn 11058 ndxcnx 15901 sSet csts 15902 Slot cslot 15903 Basecbs 15904 ↾s cress 15905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-nn 11059 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 |
This theorem is referenced by: ressplusg 16040 ressmulr 16053 ressstarv 16054 resssca 16078 ressvsca 16079 ressip 16080 resstset 16093 ressle 16106 ressds 16120 resshom 16125 ressco 16126 ressunif 22113 |
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