Theorem bj-pr11val 35195

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

Assertion

Ref	Expression
bj-pr11val	⊢ pr1 ⦅𝐴⦆ = 𝐴

Proof of Theorem bj-pr11val

Step	Hyp	Ref	Expression
1		df-bj-1upl 35188	. . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
2		bj-pr1eq 35192	. . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)
4		bj-pr1val 35194	. 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅)
5		eqid 2738	. . 3 ⊢ ∅ = ∅
6	5	iftruei 4466	. 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴
7	3, 4, 6	3eqtri 2770	1 ⊢ pr1 ⦅𝐴⦆ = 𝐴

Syntax hints:   = wceq 1539  ∅c0 4256  ifcif 4459  {csn 4561   × cxp 5587  tag bj-ctag 35164  ⦅bj-c1upl 35187  pr₁ bj-cpr1 35190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-bj-sngl 35156  df-bj-tag 35165  df-bj-proj 35181  df-bj-1upl 35188  df-bj-pr1 35191

This theorem is referenced by:  bj-1uplth  35197  bj-1uplex  35198  bj-pr21val  35203