Theorem bj-pr1val 32992

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

Assertion

Ref	Expression
bj-pr1val	|- pr1 ( { A } X. tag B ) = if ( A = (/) , B , (/) )

Proof of Theorem bj-pr1val

Step	Hyp	Ref	Expression
1		df-bj-pr1 32989	. 2 |- pr1 ( { A } X. tag B ) = ( (/) Proj ( { A } X. tag B ) )
2		0ex 4790	. . 3 |- (/) e. _V
3		bj-projval 32984	. . 3 |- ( (/) e. _V -> ( (/) Proj ( { A } X. tag B ) ) = if ( A = (/) , B , (/) ) )
4	2, 3	ax-mp 5	. 2 |- ( (/) Proj ( { A } X. tag B ) ) = if ( A = (/) , B , (/) )
5	1, 4	eqtri 2644	1 |- pr1 ( { A } X. tag B ) = if ( A = (/) , B , (/) )

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   X. cxp 5112  tag bj-ctag 32962   Proj bj-cproj 32978  pr₁ bj-cpr1 32988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-bj-sngl 32954  df-bj-tag 32963  df-bj-proj 32979  df-bj-pr1 32989

This theorem is referenced by:  bj-pr11val  32993  bj-pr21val  33001