Theorem bj-pr11val 36993

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 36986	. . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
2		bj-pr1eq 36990	. . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)
4		bj-pr1val 36992	. 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅)
5		eqid 2729	. . 3 ⊢ ∅ = ∅
6	5	iftruei 4495	. 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴
7	3, 4, 6	3eqtri 2756	1 ⊢ pr1 ⦅𝐴⦆ = 𝐴

Syntax hints:   = wceq 1540  ∅c0 4296  ifcif 4488  {csn 4589   × cxp 5636  tag bj-ctag 36962  ⦅bj-c1upl 36985  pr₁ bj-cpr1 36988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-bj-sngl 36954  df-bj-tag 36963  df-bj-proj 36979  df-bj-1upl 36986  df-bj-pr1 36989

This theorem is referenced by:  bj-1uplth  36995  bj-1uplex  36996  bj-pr21val  37001