Theorem bj-pr2val 37344

Description: Value of the second projection. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-pr2val	⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)

Proof of Theorem bj-pr2val

Step	Hyp	Ref	Expression
1		df-bj-pr2 37341	. 2 ⊢ pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵))
2		1oex 8409	. . 3 ⊢ 1o ∈ V
3		bj-projval 37322	. . 3 ⊢ (1o ∈ V → (1o Proj ({𝐴} × tag 𝐵)) = if(𝐴 = 1o, 𝐵, ∅))
4	2, 3	ax-mp 5	. 2 ⊢ (1o Proj ({𝐴} × tag 𝐵)) = if(𝐴 = 1o, 𝐵, ∅)
5	1, 4	eqtri 2760	1 ⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  ifcif 4467  {csn 4568   × cxp 5623  1_oc1o 8392  tag bj-ctag 37300   Proj bj-cproj 37316  pr₂ bj-cpr2 37340

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6324  df-1o 8399  df-bj-sngl 37292  df-bj-tag 37301  df-bj-proj 37317  df-bj-pr2 37341

This theorem is referenced by:  bj-pr22val  37345