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Mirrors > Home > ILE Home > Th. List > setsabsd | GIF version |
Description: Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.) |
Ref | Expression |
---|---|
setsabsd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
setsabsd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
setsabsd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
setsabsd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑈) |
Ref | Expression |
---|---|
setsabsd | ⊢ (𝜑 → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (𝑆 sSet 〈𝐴, 𝐶〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsabsd.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
2 | setsabsd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
3 | setsabsd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
4 | setsresg 12002 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) | |
5 | 1, 2, 3, 4 | syl3anc 1216 | . . 3 ⊢ (𝜑 → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) |
6 | 5 | uneq1d 3229 | . 2 ⊢ (𝜑 → (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
7 | setsex 11996 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) | |
8 | 1, 2, 3, 7 | syl3anc 1216 | . . 3 ⊢ (𝜑 → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) |
9 | setsabsd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
10 | setsvala 11995 | . . 3 ⊢ (((𝑆 sSet 〈𝐴, 𝐵〉) ∈ V ∧ 𝐴 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) | |
11 | 8, 2, 9, 10 | syl3anc 1216 | . 2 ⊢ (𝜑 → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
12 | setsvala 11995 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → (𝑆 sSet 〈𝐴, 𝐶〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) | |
13 | 1, 2, 9, 12 | syl3anc 1216 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈𝐴, 𝐶〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐶〉})) |
14 | 6, 11, 13 | 3eqtr4d 2182 | 1 ⊢ (𝜑 → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (𝑆 sSet 〈𝐴, 𝐶〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∖ cdif 3068 ∪ cun 3069 {csn 3527 〈cop 3530 ↾ cres 4541 (class class class)co 5774 sSet csts 11962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sets 11971 |
This theorem is referenced by: (None) |
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