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Theorem bj-projval 36181
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 4643 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 bj-xpima2sn 36143 . . . . . . . . 9 (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
31, 2syl6bir 253 . . . . . . . 8 (𝐴𝑉 → (𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶))
43imp 406 . . . . . . 7 ((𝐴𝑉𝐴 = 𝐵) → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
54eleq2d 2818 . . . . . 6 ((𝐴𝑉𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
65abbidv 2800 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7 df-bj-proj 36176 . . . . 5 (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8 bj-taginv 36171 . . . . 5 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
96, 7, 83eqtr4g 2796 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶)
109ex 412 . . 3 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶))
11 noel 4331 . . . . 5 ¬ {𝑥} ∈ ∅
127eqabri 2876 . . . . . 6 (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13 elsni 4646 . . . . . . . 8 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14 bj-xpima1sn 36141 . . . . . . . 8 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1513, 14nsyl5 159 . . . . . . 7 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1615eleq2d 2818 . . . . . 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 4405 . . 3 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20 ifval 4571 . . 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 589 . 2 (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅))
22 eqcom 2738 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
23 ifbi 4551 . . 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 2787 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 395   = wceq 1540  wcel 2105  {cab 2708  c0 4323  ifcif 4529  {csn 4629   × cxp 5675  cima 5680  tag bj-ctag 36159   Proj bj-cproj 36175
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-bj-sngl 36151  df-bj-tag 36160  df-bj-proj 36176
This theorem is referenced by:  bj-pr1val  36189  bj-pr2val  36203
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