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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 17140 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) | |
2 | 1 | adantr 480 | . . 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 7449 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V) | |
5 | setsval 17129 | . . 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 579 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩})) |
7 | setsval 17129 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐶⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩})) | |
8 | 3, 6, 7 | 3eqtr4d 2778 | 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 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∖ cdif 3942 ∪ cun 3943 {csn 4624 ⟨cop 4630 ↾ cres 5674 (class class class)co 7414 sSet csts 17125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-res 5684 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-sets 17126 |
This theorem is referenced by: ressress 17222 rescabs 17811 rescabsOLD 17812 opprabs 33187 |
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