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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 3461 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ V) |
3 | setsnidel.n | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
4 | 3 | necomd 3042 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
5 | eldifsn 4680 | . . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴)) | |
6 | 2, 4, 5 | sylanbrc 586 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴})) |
7 | setsnidel.s | . . . 4 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑆) | |
8 | setsnidel.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
9 | opelres 5824 | . . . . 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 712 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴}))) |
12 | elun1 4103 | . . 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 16505 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
18 | 15, 16, 17 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
19 | 14, 18 | syl5eq 2845 | . 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
20 | 13, 19 | eleqtrrd 2893 | 1 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 {csn 4525 〈cop 4531 ↾ cres 5521 (class class class)co 7135 sSet csts 16473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-sets 16482 |
This theorem is referenced by: (None) |
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