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Mirrors > Home > MPE Home > Th. List > Mathboxes > setsidel | Structured version Visualization version GIF version |
Description: The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.) |
Ref | Expression |
---|---|
setsidel.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
setsidel.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
setsidel.r | ⊢ 𝑅 = (𝑆 sSet 〈𝐴, 𝐵〉) |
Ref | Expression |
---|---|
setsidel | ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5383 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | 1 | snid 4603 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} |
3 | elun2 4116 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} → 〈𝐴, 𝐵〉 ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
5 | setsidel.r | . . 3 ⊢ 𝑅 = (𝑆 sSet 〈𝐴, 𝐵〉) | |
6 | setsidel.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
7 | setsidel.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
8 | setsval 16866 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
9 | 6, 7, 8 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
10 | 5, 9 | eqtrid 2792 | . 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
11 | 4, 10 | eleqtrrd 2844 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∖ cdif 3889 ∪ cun 3890 {csn 4567 〈cop 4573 ↾ cres 5592 (class class class)co 7271 sSet csts 16862 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-res 5602 df-iota 6390 df-fun 6434 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-sets 16863 |
This theorem is referenced by: (None) |
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