Theorem bj-projval 36962

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 4662	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		bj-xpima2sn 36924	. . . . . . . . 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 2830	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
6	5	abbidv 2811	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7		df-bj-proj 36957	. . . . 5 ⊢ (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8		bj-taginv 36952	. . . . 5 ⊢ 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
9	6, 7, 8	3eqtr4g 2805	. . . 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 4360	. . . . 5 ⊢ ¬ {𝑥} ∈ ∅
12	7	eqabri 2888	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13		elsni 4665	. . . . . . . 8 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14		bj-xpima1sn 36922	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
15	13, 14	nsyl5 159	. . . . . . 7 ⊢ (¬ 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
16	15	eleq2d 2830	. . . . . 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 4430	. . 3 ⊢ (¬ 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20		ifval 4590	. . 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 2747	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
23		ifbi 4570	. . 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 2796	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∅c0 4352  ifcif 4548  {csn 4648   × cxp 5698   “ cima 5703  tag bj-ctag 36940   Proj bj-cproj 36956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-bj-sngl 36932  df-bj-tag 36941  df-bj-proj 36957

This theorem is referenced by:  bj-pr1val  36970  bj-pr2val  36984