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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 3513 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
2 | elex 3513 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
3 | resexg 5893 | . . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V) | |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V) |
5 | snex 5324 | . . . 4 ⊢ {𝐴} ∈ V | |
6 | unexg 7466 | . . . 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 4573 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → {𝑒} = {𝐴}) |
11 | 10 | dmeqd 5769 | . . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → dom {𝑒} = dom {𝐴}) |
12 | 11 | difeq2d 4099 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → (V ∖ dom {𝑒}) = (V ∖ dom {𝐴})) |
13 | 8, 12 | reseq12d 5849 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → (𝑠 ↾ (V ∖ dom {𝑒})) = (𝑆 ↾ (V ∖ dom {𝐴}))) |
14 | 13, 10 | uneq12d 4140 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
15 | df-sets 16484 | . . . 4 ⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) | |
16 | 14, 15 | ovmpoga 7298 | . . 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 1533 ∈ wcel 2110 Vcvv 3495 ∖ cdif 3933 ∪ cun 3934 {csn 4561 dom cdm 5550 ↾ cres 5552 (class class class)co 7150 sSet csts 16475 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
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 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-res 5562 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-sets 16484 |
This theorem is referenced by: setsval 16507 setsdm 16511 setsfun 16512 setsfun0 16513 wunsets 16518 setsres 16519 |
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