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Mirrors > Home > MPE Home > Th. List > setsvalg | Structured version Visualization version GIF version |
Description: Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
setsvalg | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3514 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
2 | elex 3514 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
3 | resexg 5900 | . . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V) | |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V) |
5 | snex 5334 | . . . 4 ⊢ {𝐴} ∈ V | |
6 | unexg 7474 | . . . 4 ⊢ (((𝑆 ↾ (V ∖ dom {𝐴})) ∈ V ∧ {𝐴} ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) | |
7 | 4, 5, 6 | sylancl 588 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) |
8 | simpl 485 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → 𝑠 = 𝑆) | |
9 | simpr 487 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → 𝑒 = 𝐴) | |
10 | 9 | sneqd 4581 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → {𝑒} = {𝐴}) |
11 | 10 | dmeqd 5776 | . . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → dom {𝑒} = dom {𝐴}) |
12 | 11 | difeq2d 4101 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → (V ∖ dom {𝑒}) = (V ∖ dom {𝐴})) |
13 | 8, 12 | reseq12d 5856 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → (𝑠 ↾ (V ∖ dom {𝑒})) = (𝑆 ↾ (V ∖ dom {𝐴}))) |
14 | 13, 10 | uneq12d 4142 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
15 | df-sets 16492 | . . . 4 ⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) | |
16 | 14, 15 | ovmpoga 7306 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V ∧ ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
17 | 7, 16 | mpd3an3 1458 | . 2 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
18 | 1, 2, 17 | syl2an 597 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 {csn 4569 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: setsval 16515 setsdm 16519 setsfun 16520 setsfun0 16521 wunsets 16526 setsres 16527 |
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