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Theorem bj-projval 36180
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 4641 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 bj-xpima2sn 36142 . . . . . . . . 9 (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
31, 2syl6bir 253 . . . . . . . 8 (𝐴𝑉 → (𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶))
43imp 405 . . . . . . 7 ((𝐴𝑉𝐴 = 𝐵) → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
54eleq2d 2817 . . . . . 6 ((𝐴𝑉𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
65abbidv 2799 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7 df-bj-proj 36175 . . . . 5 (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8 bj-taginv 36170 . . . . 5 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
96, 7, 83eqtr4g 2795 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶)
109ex 411 . . 3 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶))
11 noel 4329 . . . . 5 ¬ {𝑥} ∈ ∅
127eqabri 2875 . . . . . 6 (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13 elsni 4644 . . . . . . . 8 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14 bj-xpima1sn 36140 . . . . . . . 8 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1513, 14nsyl5 159 . . . . . . 7 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1615eleq2d 2817 . . . . . 6 𝐴 = 𝐵 → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ∅))
1712, 16bitrid 282 . . . . 5 𝐴 = 𝐵 → (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ ∅))
1811, 17mtbiri 326 . . . 4 𝐴 = 𝐵 → ¬ 𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)))
1918eq0rdv 4403 . . 3 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20 ifval 4569 . . 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)) = ∅)))
2110, 19, 20sylanblrc 588 . 2 (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅))
22 eqcom 2737 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
23 ifbi 4549 . . 3 ((𝐴 = 𝐵𝐵 = 𝐴) → if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅))
2422, 23ax-mp 5 . 2 if(𝐴 = 𝐵, 𝐶, ∅) = if(𝐵 = 𝐴, 𝐶, ∅)
2521, 24eqtrdi 2786 1 (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  {cab 2707  c0 4321  ifcif 4527  {csn 4627   × cxp 5673  cima 5678  tag bj-ctag 36158   Proj bj-cproj 36174
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-bj-sngl 36150  df-bj-tag 36159  df-bj-proj 36175
This theorem is referenced by:  bj-pr1val  36188  bj-pr2val  36202
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