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Theorem bj-projval 33934
 Description: Value of the class projection. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-projval (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Proof of Theorem bj-projval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elsng 4492 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 bj-xpima2sn 33847 . . . . . . . . 9 (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
31, 2syl6bir 255 . . . . . . . 8 (𝐴𝑉 → (𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶))
43imp 407 . . . . . . 7 ((𝐴𝑉𝐴 = 𝐵) → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
54eleq2d 2870 . . . . . 6 ((𝐴𝑉𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
65abbidv 2862 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7 df-bj-proj 33929 . . . . 5 (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8 bj-taginv 33924 . . . . 5 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
96, 7, 83eqtr4g 2858 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶)
109ex 413 . . 3 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶))
11 noel 4222 . . . . 5 ¬ {𝑥} ∈ ∅
127abeq2i 2919 . . . . . 6 (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13 elsni 4495 . . . . . . . . 9 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
1413con3i 157 . . . . . . . 8 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵})
15 bj-xpima1sn 33845 . . . . . . . 8 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1614, 15syl 17 . . . . . . 7 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1716eleq2d 2870 . . . . . 6 𝐴 = 𝐵 → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ∅))
1812, 17syl5bb 284 . . . . 5 𝐴 = 𝐵 → (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ ∅))
1911, 18mtbiri 328 . . . 4 𝐴 = 𝐵 → ¬ 𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)))
2019eq0rdv 4283 . . 3 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
21 ifval 4428 . . 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)) = ∅)))
2210, 20, 21sylanblrc 590 . 2 (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅))
23 eqcom 2804 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
24 ifbi 4408 . . 3 ((𝐴 = 𝐵𝐵 = 𝐴) → if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅))
2523, 24ax-mp 5 . 2 if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅)
2622, 25syl6eq 2849 1 (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  {cab 2777  ∅c0 4217  ifcif 4387  {csn 4478   × cxp 5448   “ cima 5453  tag bj-ctag 33912   Proj bj-cproj 33928 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-bj-sngl 33904  df-bj-tag 33913  df-bj-proj 33929 This theorem is referenced by:  bj-pr1val  33942  bj-pr2val  33956
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