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Mirrors > Home > MPE Home > Th. List > Mathboxes > setsnidel | 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〉) |
setsnidel.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
setsnidel.d | ⊢ (𝜑 → 𝐷 ∈ 𝑌) |
setsnidel.s | ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑆) |
setsnidel.n | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Ref | Expression |
---|---|
setsnidel | ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsnidel.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
2 | 1 | elexd 3514 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ V) |
3 | setsnidel.n | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
4 | 3 | necomd 3071 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
5 | eldifsn 4712 | . . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴)) | |
6 | 2, 4, 5 | sylanbrc 585 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴})) |
7 | setsnidel.s | . . . 4 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑆) | |
8 | setsnidel.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
9 | opelres 5853 | . . . . 5 ⊢ (𝐷 ∈ 𝑌 → (〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ 〈𝐶, 𝐷〉 ∈ 𝑆))) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ 〈𝐶, 𝐷〉 ∈ 𝑆))) |
11 | 6, 7, 10 | mpbir2and 711 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴}))) |
12 | elun1 4151 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴})) → 〈𝐶, 𝐷〉 ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
14 | setsidel.r | . . 3 ⊢ 𝑅 = (𝑆 sSet 〈𝐴, 𝐵〉) | |
15 | setsidel.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
16 | setsidel.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
17 | setsval 16507 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
18 | 15, 16, 17 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
19 | 14, 18 | syl5eq 2868 | . 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
20 | 13, 19 | eleqtrrd 2916 | 1 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ∪ cun 3933 {csn 4560 〈cop 4566 ↾ cres 5551 (class class class)co 7150 sSet csts 16475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-sets 16484 |
This theorem is referenced by: (None) |
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