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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 2610 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | 1, 2 | ressid2 15755 | . . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
4 | 3 | fveq2d 6107 | . . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
5 | 4 | 3expib 1260 | . . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
6 | 1, 2 | ressval2 15756 | . . . . . . 7 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
7 | 6 | fveq2d 6107 | . . . . . 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 15716 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
11 | 8, 9 | ndxarg 15715 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
12 | 1re 9918 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
13 | resslem.b | . . . . . . . . . 10 ⊢ 1 < 𝑁 | |
14 | 12, 13 | gtneii 10028 | . . . . . . . . 9 ⊢ 𝑁 ≠ 1 |
15 | 11, 14 | eqnetri 2852 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 1 |
16 | basendx 15751 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
17 | 15, 16 | neeqtrri 2855 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Base‘ndx) |
18 | 10, 17 | setsnid 15743 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
19 | 7, 18 | syl6eqr 2662 | . . . . 5 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
20 | 19 | 3expib 1260 | . . . 4 ⊢ (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
21 | 5, 20 | pm2.61i 175 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
22 | reldmress 15753 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
23 | 22 | ovprc1 6582 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
24 | 1, 23 | syl5eq 2656 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
25 | 24 | fveq2d 6107 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
26 | 8 | str0 15739 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
27 | 25, 26 | syl6eqr 2662 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
28 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
29 | 27, 28 | eqtr4d 2647 | . . . 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 827 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
32 | resslem.e | . 2 ⊢ 𝐶 = (𝐸‘𝑊) | |
33 | 31, 32 | syl6reqr 2663 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 1c1 9816 < clt 9953 ℕcn 10897 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 Basecbs 15695 ↾s cress 15696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 |
This theorem is referenced by: ressplusg 15818 ressmulr 15829 ressstarv 15830 resssca 15854 ressvsca 15855 ressip 15856 resstset 15869 ressle 15882 ressds 15896 resshom 15901 ressco 15902 ressunif 21876 |
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