Theorem bj-projval 34311

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 4581	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2		bj-xpima2sn 34273	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
3	1, 2	syl6bir 256	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶))
4	3	imp 409	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
5	4	eleq2d 2898	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
6	5	abbidv 2885	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7		df-bj-proj 34306	. . . . 5 ⊢ (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8		bj-taginv 34301	. . . . 5 ⊢ 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
9	6, 7, 8	3eqtr4g 2881	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶)
10	9	ex 415	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶))
11		noel 4296	. . . . 5 ⊢ ¬ {𝑥} ∈ ∅
12	7	abeq2i 2948	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13		elsni 4584	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14	13	con3i 157	. . . . . . . 8 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵})
15		bj-xpima1sn 34271	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
16	14, 15	syl 17	. . . . . . 7 ⊢ (¬ 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
17	16	eleq2d 2898	. . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ∅))
18	12, 17	syl5bb 285	. . . . 5 ⊢ (¬ 𝐴 = 𝐵 → (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ ∅))
19	11, 18	mtbiri 329	. . . 4 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)))
20	19	eq0rdv 4357	. . 3 ⊢ (¬ 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
21		ifval 4508	. . 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)) = ∅)))
22	10, 20, 21	sylanblrc 592	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅))
23		eqcom 2828	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
24		ifbi 4488	. . 3 ⊢ ((𝐴 = 𝐵 ↔ 𝐵 = 𝐴) → if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅))
25	23, 24	ax-mp 5	. 2 ⊢ if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅)
26	22, 25	syl6eq 2872	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799  ∅c0 4291  ifcif 4467  {csn 4567   × cxp 5553   “ cima 5558  tag bj-ctag 34289   Proj bj-cproj 34305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-bj-sngl 34281  df-bj-tag 34290  df-bj-proj 34306

This theorem is referenced by:  bj-pr1val  34319  bj-pr2val  34333