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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 2728 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | ressid2 17212 | . . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
5 | 4 | fveq2d 6901 | . . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
6 | 5 | 3expib 1120 | . . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
7 | 2, 3 | ressval2 17213 | . . . . . . 7 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
8 | 7 | fveq2d 6901 | . . . . . 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 17177 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
12 | 8, 11 | eqtr4di 2786 | . . . . 5 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
13 | 12 | 3expib 1120 | . . . 4 ⊢ (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
14 | 6, 13 | pm2.61i 182 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
15 | 9 | str0 17157 | . . . . . . 7 ⊢ ∅ = (𝐸‘∅) |
16 | 15 | eqcomi 2737 | . . . . . 6 ⊢ (𝐸‘∅) = ∅ |
17 | reldmress 17210 | . . . . . 6 ⊢ Rel dom ↾s | |
18 | 16, 2, 17 | oveqprc 17160 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘𝑅)) |
19 | 18 | eqcomd 2734 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
20 | 19 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
21 | 14, 20 | pm2.61ian 811 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
22 | 1, 21 | eqtr4id 2787 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 〈cop 4635 ‘cfv 6548 (class class class)co 7420 sSet csts 17131 Slot cslot 17149 ndxcnx 17161 Basecbs 17179 ↾s cress 17208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-res 5690 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-sets 17132 df-slot 17150 df-ress 17209 |
This theorem is referenced by: ressplusg 17270 ressmulr 17287 ressstarv 17288 resssca 17323 ressvsca 17324 ressip 17325 resstset 17345 ressle 17360 ressunif 17382 ressds 17390 resshom 17399 ressco 17400 |
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