Theorem bj-projval 36984

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 4603	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		bj-xpima2sn 36946	. . . . . . . . 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 2814	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
6	5	abbidv 2795	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7		df-bj-proj 36979	. . . . 5 ⊢ (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8		bj-taginv 36974	. . . . 5 ⊢ 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
9	6, 7, 8	3eqtr4g 2789	. . . 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 4301	. . . . 5 ⊢ ¬ {𝑥} ∈ ∅
12	7	eqabri 2871	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13		elsni 4606	. . . . . . . 8 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14		bj-xpima1sn 36944	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
15	13, 14	nsyl5 159	. . . . . . 7 ⊢ (¬ 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
16	15	eleq2d 2814	. . . . . 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 4370	. . 3 ⊢ (¬ 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20		ifval 4531	. . 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 2736	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
23		ifbi 4511	. . 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 2780	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 2109  {cab 2707  ∅c0 4296  ifcif 4488  {csn 4589   × cxp 5636   “ cima 5641  tag bj-ctag 36962   Proj bj-cproj 36978

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-bj-sngl 36954  df-bj-tag 36963  df-bj-proj 36979

This theorem is referenced by:  bj-pr1val  36992  bj-pr2val  37006