Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5358 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | setsvalg 16514 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ V) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) |
4 | dmsnopg 6072 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
5 | 4 | difeq2d 4101 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (V ∖ dom {〈𝐴, 𝐵〉}) = (V ∖ {𝐴})) |
6 | 5 | reseq2d 5855 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) = (𝑆 ↾ (V ∖ {𝐴}))) |
7 | 6 | uneq1d 4140 | . 2 ⊢ (𝐵 ∈ 𝑊 → ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
8 | 3, 7 | sylan9eq 2878 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 {csn 4569 〈cop 4575 dom cdm 5557 ↾ cres 5559 (class class class)co 7158 sSet csts 16483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-sets 16492 |
This theorem is referenced by: setsidvald 16516 fvsetsid 16517 fsets 16518 setsabs 16528 setscom 16529 setsid 16540 estrres 17391 symgvalstruct 18527 setsstrset 34430 setsidel 43543 setsnidel 43544 setsv 43545 |
Copyright terms: Public domain | W3C validator |