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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 16628 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) |
3 | 2 | uneq1d 4052 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
4 | ovexd 7205 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) | |
5 | setsval 16616 | . . 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 583 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
7 | setsval 16616 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐶〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) | |
8 | 3, 6, 7 | 3eqtr4d 2783 | 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 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∖ cdif 3840 ∪ cun 3841 {csn 4516 〈cop 4522 ↾ cres 5527 (class class class)co 7170 sSet csts 16584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-res 5537 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-sets 16593 |
This theorem is referenced by: ressress 16665 rescabs 17208 |
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