Theorem bj-projval 35113

Description: Value of the class projection. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-projval	⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Proof of Theorem bj-projval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsng 4572	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		bj-xpima2sn 35075	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
3	1, 2	syl6bir 253	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶))
4	3	imp 406	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
5	4	eleq2d 2824	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
6	5	abbidv 2808	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7		df-bj-proj 35108	. . . . 5 ⊢ (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8		bj-taginv 35103	. . . . 5 ⊢ 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
9	6, 7, 8	3eqtr4g 2804	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶)
10	9	ex 412	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶))
11		noel 4261	. . . . 5 ⊢ ¬ {𝑥} ∈ ∅
12	7	abeq2i 2874	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13		elsni 4575	. . . . . . . 8 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14		bj-xpima1sn 35073	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
15	13, 14	nsyl5 159	. . . . . . 7 ⊢ (¬ 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
16	15	eleq2d 2824	. . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ∅))
17	12, 16	syl5bb 282	. . . . 5 ⊢ (¬ 𝐴 = 𝐵 → (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ ∅))
18	11, 17	mtbiri 326	. . . 4 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)))
19	18	eq0rdv 4335	. . 3 ⊢ (¬ 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20		ifval 4498	. . 3 ⊢ ((𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅) ↔ ((𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶) ∧ (¬ 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)))
21	10, 19, 20	sylanblrc 589	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅))
22		eqcom 2745	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
23		ifbi 4478	. . 3 ⊢ ((𝐴 = 𝐵 ↔ 𝐵 = 𝐴) → if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅))
24	22, 23	ax-mp 5	. 2 ⊢ if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅)
25	21, 24	eqtrdi 2795	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∅c0 4253  ifcif 4456  {csn 4558   × cxp 5578   “ cima 5583  tag bj-ctag 35091   Proj bj-cproj 35107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-bj-sngl 35083  df-bj-tag 35092  df-bj-proj 35108

This theorem is referenced by:  bj-pr1val  35121  bj-pr2val  35135