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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 3455 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
2 | elex 3455 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
3 | resexg 5779 | . . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V) |
5 | snex 5223 | . . . 4 ⊢ {𝐴} ∈ V | |
6 | unexg 7329 | . . . 4 ⊢ (((𝑆 ↾ (V ∖ dom {𝐴})) ∈ V ∧ {𝐴} ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) | |
7 | 4, 5, 6 | sylancl 586 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) |
8 | simpl 483 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → 𝑠 = 𝑆) | |
9 | simpr 485 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → 𝑒 = 𝐴) | |
10 | 9 | sneqd 4484 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → {𝑒} = {𝐴}) |
11 | 10 | dmeqd 5660 | . . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → dom {𝑒} = dom {𝐴}) |
12 | 11 | difeq2d 4020 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → (V ∖ dom {𝑒}) = (V ∖ dom {𝐴})) |
13 | 8, 12 | reseq12d 5735 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → (𝑠 ↾ (V ∖ dom {𝑒})) = (𝑆 ↾ (V ∖ dom {𝐴}))) |
14 | 13, 10 | uneq12d 4061 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑒 = 𝐴) → ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
15 | df-sets 16319 | . . . 4 ⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) | |
16 | 14, 15 | ovmpoga 7160 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V ∧ ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
17 | 7, 16 | mpd3an3 1454 | . 2 ⊢ ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
18 | 1, 2, 17 | syl2an 595 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∖ cdif 3856 ∪ cun 3857 {csn 4472 dom cdm 5443 ↾ cres 5445 (class class class)co 7016 sSet csts 16310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-res 5455 df-iota 6189 df-fun 6227 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-sets 16319 |
This theorem is referenced by: setsval 16342 setsdm 16346 setsfun 16347 setsfun0 16348 wunsets 16353 setsres 16354 |
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