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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 4246 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
5 | 3, 4 | eqsstrrdi 4051 | . 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 2787 | . . . 4 ⊢ (¬ 𝐴 ∈ V → 𝑅 = ∅) |
9 | 8 | fveq2d 6911 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Base‘𝑅) = (Base‘∅)) |
10 | base0 17250 | . . . 4 ⊢ ∅ = (Base‘∅) | |
11 | 0ss 4406 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
12 | 10, 11 | eqsstrri 4031 | . . 3 ⊢ (Base‘∅) ⊆ 𝐵 |
13 | 9, 12 | eqsstrdi 4050 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 |
This theorem is referenced by: (None) |
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