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Theorem bj-projval 36703
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 4647 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 bj-xpima2sn 36665 . . . . . . . . 9 (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
31, 2biimtrrdi 253 . . . . . . . 8 (𝐴𝑉 → (𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶))
43imp 405 . . . . . . 7 ((𝐴𝑉𝐴 = 𝐵) → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
54eleq2d 2812 . . . . . 6 ((𝐴𝑉𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
65abbidv 2795 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7 df-bj-proj 36698 . . . . 5 (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8 bj-taginv 36693 . . . . 5 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
96, 7, 83eqtr4g 2791 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶)
109ex 411 . . 3 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶))
11 noel 4333 . . . . 5 ¬ {𝑥} ∈ ∅
127eqabri 2870 . . . . . 6 (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13 elsni 4650 . . . . . . . 8 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14 bj-xpima1sn 36663 . . . . . . . 8 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1513, 14nsyl5 159 . . . . . . 7 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1615eleq2d 2812 . . . . . 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 4409 . . 3 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20 ifval 4575 . . 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 2733 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
23 ifbi 4555 . . 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 2782 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 1534  wcel 2099  {cab 2703  c0 4325  ifcif 4533  {csn 4633   × cxp 5680  cima 5685  tag bj-ctag 36681   Proj bj-cproj 36697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-bj-sngl 36673  df-bj-tag 36682  df-bj-proj 36698
This theorem is referenced by:  bj-pr1val  36711  bj-pr2val  36725
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