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Mirrors > Home > MPE Home > Th. List > resseqnbas | Structured version Visualization version GIF version |
Description: The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.) |
Ref | Expression |
---|---|
resseqnbas.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
resseqnbas.e | ⊢ 𝐶 = (𝐸‘𝑊) |
resseqnbas.f | ⊢ 𝐸 = Slot (𝐸‘ndx) |
resseqnbas.n | ⊢ (𝐸‘ndx) ≠ (Base‘ndx) |
Ref | Expression |
---|---|
resseqnbas | ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resseqnbas.e | . 2 ⊢ 𝐶 = (𝐸‘𝑊) | |
2 | resseqnbas.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
3 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | ressid2 16788 | . . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
5 | 4 | fveq2d 6721 | . . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
6 | 5 | 3expib 1124 | . . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
7 | 2, 3 | ressval2 16789 | . . . . . . 7 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
8 | 7 | fveq2d 6721 | . . . . . 6 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
9 | resseqnbas.f | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
10 | resseqnbas.n | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Base‘ndx) | |
11 | 9, 10 | setsnid 16759 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
12 | 8, 11 | eqtr4di 2796 | . . . . 5 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
13 | 12 | 3expib 1124 | . . . 4 ⊢ (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
14 | 6, 13 | pm2.61i 185 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
15 | reldmress 16786 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
16 | 15 | ovprc1 7252 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
17 | 2, 16 | syl5eq 2790 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
18 | 17 | fveq2d 6721 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
19 | 9 | str0 16742 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
20 | 18, 19 | eqtr4di 2796 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
21 | fvprc 6709 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
22 | 20, 21 | eqtr4d 2780 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
23 | 22 | adantr 484 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
24 | 14, 23 | pm2.61ian 812 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
25 | 1, 24 | eqtr4id 2797 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 ∅c0 4237 〈cop 4547 ‘cfv 6380 (class class class)co 7213 sSet csts 16716 Slot cslot 16734 ndxcnx 16744 Basecbs 16760 ↾s cress 16784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-res 5563 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-sets 16717 df-slot 16735 df-ress 16785 |
This theorem is referenced by: ressplusg 16834 ressmulr 16848 ressstarv 16849 resssca 16876 ressvsca 16877 ressip 16878 resstset 16892 ressle 16900 ressunif 16909 ressds 16917 resshom 16923 ressco 16924 |
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