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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 5469 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 2 | 1 | snid 4662 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} | 
| 3 | elun2 4183 | . . 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 17204 | . . . 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 2789 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 {csn 4626 〈cop 4632 ↾ cres 5687 (class class class)co 7431 sSet csts 17200 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 | 
| This theorem is referenced by: (None) | 
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