Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 3512 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ V) |
3 | setsnidel.n | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
4 | 3 | necomd 3068 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
5 | eldifsn 4711 | . . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴)) | |
6 | 2, 4, 5 | sylanbrc 583 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴})) |
7 | setsnidel.s | . . . 4 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑆) | |
8 | setsnidel.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
9 | opelres 5852 | . . . . 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 709 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴}))) |
12 | elun1 4149 | . . 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 16501 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
18 | 15, 16, 17 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
19 | 14, 18 | syl5eq 2865 | . 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
20 | 13, 19 | eleqtrrd 2913 | 1 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 {csn 4557 〈cop 4563 ↾ cres 5550 (class class class)co 7145 sSet csts 16469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-sets 16478 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |