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| Mirrors > Home > MPE Home > Th. List > resslemOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of resseqnbas 17287 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 2737 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 4 | 2, 3 | ressid2 17278 | . . . . . 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 1123 | . . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
| 7 | 2, 3 | ressval2 17279 | . . . . . . 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 17234 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
| 12 | 9, 10 | ndxarg 17233 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
| 13 | 1re 11261 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
| 14 | resslemOLD.b | . . . . . . . . . 10 ⊢ 1 < 𝑁 | |
| 15 | 13, 14 | gtneii 11373 | . . . . . . . . 9 ⊢ 𝑁 ≠ 1 |
| 16 | 12, 15 | eqnetri 3011 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 1 |
| 17 | basendx 17256 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
| 18 | 16, 17 | neeqtrri 3014 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Base‘ndx) |
| 19 | 11, 18 | setsnid 17245 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) |
| 20 | 8, 19 | eqtr4di 2795 | . . . . 5 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
| 21 | 20 | 3expib 1123 | . . . 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 17276 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
| 24 | 23 | ovprc1 7470 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
| 25 | 2, 24 | eqtrid 2789 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
| 26 | 25 | fveq2d 6910 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
| 27 | 9 | str0 17226 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
| 28 | 26, 27 | eqtr4di 2795 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
| 29 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
| 30 | 28, 29 | eqtr4d 2780 | . . . 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 2796 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 1c1 11156 < clt 11295 ℕcn 12266 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 Basecbs 17247 ↾s cress 17274 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulrcl 11218 ax-i2m1 11223 ax-1ne0 11224 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 |
| This theorem is referenced by: (None) |
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