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Mirrors > Home > MPE Home > Th. List > setsval | Structured version Visualization version GIF version |
Description: Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
setsval | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5153 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | setsvalg 16251 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ V) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) | |
3 | 1, 2 | mpan2 682 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) |
4 | dmsnopg 5847 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
5 | 4 | difeq2d 3955 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (V ∖ dom {〈𝐴, 𝐵〉}) = (V ∖ {𝐴})) |
6 | 5 | reseq2d 5629 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) = (𝑆 ↾ (V ∖ {𝐴}))) |
7 | 6 | uneq1d 3993 | . 2 ⊢ (𝐵 ∈ 𝑊 → ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
8 | 3, 7 | sylan9eq 2881 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∖ cdif 3795 ∪ cun 3796 {csn 4397 〈cop 4403 dom cdm 5342 ↾ cres 5344 (class class class)co 6905 sSet csts 16220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-res 5354 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-sets 16229 |
This theorem is referenced by: setsidvald 16253 fvsetsid 16254 fsets 16255 setsabs 16265 setscom 16266 setsid 16277 estrresOLD 17131 estrres 17132 setsstrset 33605 setsidel 42227 setsnidel 42228 setsv 42229 |
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