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| Mirrors > Home > MPE Home > Th. List > ressbasssOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of ressbas 17280 as of 25-Feb-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| ressbas.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
| ressbas.b | ⊢ 𝐵 = (Base‘𝑊) |
| Ref | Expression |
|---|---|
| ressbasssOLD | ⊢ (Base‘𝑅) ⊆ 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ressbas.r | . . . 4 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
| 2 | ressbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
| 3 | 1, 2 | ressbas 17280 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
| 4 | inss2 4238 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
| 5 | 3, 4 | eqsstrrdi 4029 | . 2 ⊢ (𝐴 ∈ V → (Base‘𝑅) ⊆ 𝐵) |
| 6 | reldmress 17276 | . . . . . 6 ⊢ Rel dom ↾s | |
| 7 | 6 | ovprc2 7471 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
| 8 | 1, 7 | eqtrid 2789 | . . . 4 ⊢ (¬ 𝐴 ∈ V → 𝑅 = ∅) |
| 9 | 8 | fveq2d 6910 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Base‘𝑅) = (Base‘∅)) |
| 10 | base0 17252 | . . . 4 ⊢ ∅ = (Base‘∅) | |
| 11 | 0ss 4400 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
| 12 | 10, 11 | eqsstrri 4031 | . . 3 ⊢ (Base‘∅) ⊆ 𝐵 |
| 13 | 9, 12 | eqsstrdi 4028 | . 2 ⊢ (¬ 𝐴 ∈ V → (Base‘𝑅) ⊆ 𝐵) |
| 14 | 5, 13 | pm2.61i 182 | 1 ⊢ (Base‘𝑅) ⊆ 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 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-1cn 11213 ax-addcl 11215 |
| 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-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-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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