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Mirrors > Home > MPE Home > Th. List > setsabs | Structured version Visualization version GIF version |
Description: Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
setsabs | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (𝑆 sSet 〈𝐴, 𝐶〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsres 16519 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) |
3 | 2 | uneq1d 4137 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
4 | ovexd 7185 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) | |
5 | setsval 16507 | . . 3 ⊢ (((𝑆 sSet 〈𝐴, 𝐵〉) ∈ V ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) | |
6 | 4, 5 | sylan 582 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
7 | setsval 16507 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐶〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) | |
8 | 3, 6, 7 | 3eqtr4d 2866 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (𝑆 sSet 〈𝐴, 𝐶〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ∪ cun 3933 {csn 4560 〈cop 4566 ↾ cres 5551 (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 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 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-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-sets 16484 |
This theorem is referenced by: ressress 16556 rescabs 17097 |
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