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Mirrors > Home > MPE Home > Th. List > ressid2 | Structured version Visualization version GIF version |
Description: General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
ressbas.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
ressbas.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
ressid2 | ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbas.r | . . . 4 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
2 | ressbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | ressval 16545 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
4 | iftrue 4472 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) = 𝑊) | |
5 | 3, 4 | sylan9eqr 2878 | . 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = 𝑊) |
6 | 5 | 3impb 1111 | 1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 ifcif 4466 〈cop 4566 ‘cfv 6349 (class class class)co 7150 ndxcnx 16474 sSet csts 16475 Basecbs 16477 ↾s cress 16478 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-ress 16485 |
This theorem is referenced by: ressbas 16548 resslem 16551 ress0 16552 ressid 16553 ressinbas 16554 ressress 16556 rescabs 17097 0symgefmndeq 18516 snsymgefmndeq 18517 mgpress 19244 psgnghm2 20719 resvsca 30898 extdg1id 31048 |
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