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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 17093 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) |
3 | 2 | uneq1d 4158 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
4 | ovexd 7428 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) | |
5 | setsval 17082 | . . 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 580 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
7 | setsval 17082 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐶〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) | |
8 | 3, 6, 7 | 3eqtr4d 2781 | 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∖ cdif 3941 ∪ cun 3942 {csn 4622 〈cop 4628 ↾ cres 5671 (class class class)co 7393 sSet csts 17078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6484 df-fun 6534 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-sets 17079 |
This theorem is referenced by: ressress 17175 rescabs 17764 rescabsOLD 17765 opprabs 32440 |
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