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Theorem bj-projval 34433
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 4542 . . . . . . . . 9 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
2 bj-xpima2sn 34395 . . . . . . . . 9 (𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
31, 2syl6bir 257 . . . . . . . 8 (𝐴𝑉 → (𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶))
43imp 410 . . . . . . 7 ((𝐴𝑉𝐴 = 𝐵) → (({𝐵} × tag 𝐶) “ {𝐴}) = tag 𝐶)
54eleq2d 2878 . . . . . 6 ((𝐴𝑉𝐴 = 𝐵) → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ tag 𝐶))
65abbidv 2865 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ tag 𝐶})
7 df-bj-proj 34428 . . . . 5 (𝐴 Proj ({𝐵} × tag 𝐶)) = {𝑥 ∣ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴})}
8 bj-taginv 34423 . . . . 5 𝐶 = {𝑥 ∣ {𝑥} ∈ tag 𝐶}
96, 7, 83eqtr4g 2861 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶)
109ex 416 . . 3 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = 𝐶))
11 noel 4250 . . . . 5 ¬ {𝑥} ∈ ∅
127abeq2i 2928 . . . . . 6 (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}))
13 elsni 4545 . . . . . . . 8 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
14 bj-xpima1sn 34393 . . . . . . . 8 𝐴 ∈ {𝐵} → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1513, 14nsyl5 162 . . . . . . 7 𝐴 = 𝐵 → (({𝐵} × tag 𝐶) “ {𝐴}) = ∅)
1615eleq2d 2878 . . . . . 6 𝐴 = 𝐵 → ({𝑥} ∈ (({𝐵} × tag 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ∅))
1712, 16syl5bb 286 . . . . 5 𝐴 = 𝐵 → (𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)) ↔ {𝑥} ∈ ∅))
1811, 17mtbiri 330 . . . 4 𝐴 = 𝐵 → ¬ 𝑥 ∈ (𝐴 Proj ({𝐵} × tag 𝐶)))
1918eq0rdv 4315 . . 3 𝐴 = 𝐵 → (𝐴 Proj ({𝐵} × tag 𝐶)) = ∅)
20 ifval 4469 . . 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 593 . 2 (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐴 = 𝐵, 𝐶, ∅))
22 eqcom 2808 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
23 ifbi 4449 . . 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 2852 1 (𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  c0 4246  ifcif 4428  {csn 4528   × cxp 5521  cima 5526  tag bj-ctag 34411   Proj bj-cproj 34427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-bj-sngl 34403  df-bj-tag 34412  df-bj-proj 34428
This theorem is referenced by:  bj-pr1val  34441  bj-pr2val  34455
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