Theorem bj-pr11val 33943

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 33936	. . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
2		bj-pr1eq 33940	. . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)
4		bj-pr1val 33942	. 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅)
5		eqid 2797	. . 3 ⊢ ∅ = ∅
6	5	iftruei 4394	. 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴
7	3, 4, 6	3eqtri 2825	1 ⊢ pr1 ⦅𝐴⦆ = 𝐴

Syntax hints:   = wceq 1525  ∅c0 4217  ifcif 4387  {csn 4478   × cxp 5448  tag bj-ctag 33912  ⦅bj-c1upl 33935  pr₁ bj-cpr1 33938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-bj-sngl 33904  df-bj-tag 33913  df-bj-proj 33929  df-bj-1upl 33936  df-bj-pr1 33939

This theorem is referenced by:  bj-1uplth  33945  bj-1uplex  33946  bj-pr21val  33951