Theorem bj-projval 37014

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 4615	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		bj-xpima2sn 36976	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
3	1, 2	biimtrrdi 254	. . . . . . . 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 2820	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
6	5	abbidv 2801	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7		df-bj-proj 37009	. . . . 5 ⊢ (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8		bj-taginv 37004	. . . . 5 ⊢ 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
9	6, 7, 8	3eqtr4g 2795	. . . 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 4313	. . . . 5 ⊢ ¬ {𝑥} ∈ ∅
12	7	eqabri 2878	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13		elsni 4618	. . . . . . . 8 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14		bj-xpima1sn 36974	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
15	13, 14	nsyl5 159	. . . . . . 7 ⊢ (¬ 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
16	15	eleq2d 2820	. . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ∅))
17	12, 16	bitrid 283	. . . . 5 ⊢ (¬ 𝐴 = 𝐵 → (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ ∅))
18	11, 17	mtbiri 327	. . . 4 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)))
19	18	eq0rdv 4382	. . 3 ⊢ (¬ 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20		ifval 4543	. . 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 590	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅))
22		eqcom 2742	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
23		ifbi 4523	. . 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 2786	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∅c0 4308  ifcif 4500  {csn 4601   × cxp 5652   “ cima 5657  tag bj-ctag 36992   Proj bj-cproj 37008

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-bj-sngl 36984  df-bj-tag 36993  df-bj-proj 37009

This theorem is referenced by:  bj-pr1val  37022  bj-pr2val  37036