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Mirrors > Home > MPE Home > Th. List > resslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of resseqnbas 17286 as of 21-Oct-2024. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
resslemOLD.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
resslemOLD.e | ⊢ 𝐶 = (𝐸‘𝑊) |
resslemOLD.f | ⊢ 𝐸 = Slot 𝑁 |
resslemOLD.n | ⊢ 𝑁 ∈ ℕ |
resslemOLD.b | ⊢ 1 < 𝑁 |
Ref | Expression |
---|---|
resslemOLD | ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resslemOLD.e | . 2 ⊢ 𝐶 = (𝐸‘𝑊) | |
2 | resslemOLD.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
3 | eqid 2734 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | ressid2 17277 | . . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
5 | 4 | fveq2d 6910 | . . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
6 | 5 | 3expib 1121 | . . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
7 | 2, 3 | ressval2 17278 | . . . . . . 7 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
8 | 7 | fveq2d 6910 | . . . . . 6 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
9 | resslemOLD.f | . . . . . . . 8 ⊢ 𝐸 = Slot 𝑁 | |
10 | resslemOLD.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
11 | 9, 10 | ndxid 17230 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
12 | 9, 10 | ndxarg 17229 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
13 | 1re 11258 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
14 | resslemOLD.b | . . . . . . . . . 10 ⊢ 1 < 𝑁 | |
15 | 13, 14 | gtneii 11370 | . . . . . . . . 9 ⊢ 𝑁 ≠ 1 |
16 | 12, 15 | eqnetri 3008 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 1 |
17 | basendx 17253 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
18 | 16, 17 | neeqtrri 3011 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Base‘ndx) |
19 | 11, 18 | setsnid 17242 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
20 | 8, 19 | eqtr4di 2792 | . . . . 5 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
21 | 20 | 3expib 1121 | . . . 4 ⊢ (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
22 | 6, 21 | pm2.61i 182 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
23 | reldmress 17275 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
24 | 23 | ovprc1 7469 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
25 | 2, 24 | eqtrid 2786 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
26 | 25 | fveq2d 6910 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
27 | 9 | str0 17222 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
28 | 26, 27 | eqtr4di 2792 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
29 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
30 | 28, 29 | eqtr4d 2777 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
31 | 30 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
32 | 22, 31 | pm2.61ian 812 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
33 | 1, 32 | eqtr4id 2793 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 〈cop 4636 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 1c1 11153 < clt 11292 ℕcn 12263 sSet csts 17196 Slot cslot 17214 ndxcnx 17226 Basecbs 17244 ↾s cress 17273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-mulcl 11214 ax-mulrcl 11215 ax-i2m1 11220 ax-1ne0 11221 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-nn 12264 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 |
This theorem is referenced by: (None) |
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