Theorem bj-pr21val 38053

Description: Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-pr21val	⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴

Proof of Theorem bj-pr21val

Step	Hyp	Ref	Expression
1		df-bj-2upl 38051	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr1eq 38042	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) → pr1 ⦅𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)))
3	1, 2	ax-mp 5	. 2 ⊢ pr1 ⦅𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
4		bj-pr1un 38043	. 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5		bj-pr11val 38045	. . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴
6		bj-pr1val 38044	. . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7		1n0 8506	. . . . . . 7 ⊢ 1o ≠ ∅
8	7	neii 2960	. . . . . 6 ⊢ ¬ 1o = ∅
9	8	iffalsei 4494	. . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅
10	6, 9	eqtri 2786	. . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅
11	5, 10	uneq12i 4167	. . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12		un0 4346	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
13	11, 12	eqtri 2786	. 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
14	3, 4, 13	3eqtri 2790	1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4151  ∅c0 4280  ifcif 4484  {csn 4587   × cxp 5652  1_oc1o 8480  tag bj-ctag 38014  ⦅bj-c1upl 38037  pr₁ bj-cpr1 38040  ⦅bj-c2uple 38050

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-suc 6374  df-1o 8487  df-bj-sngl 38006  df-bj-tag 38015  df-bj-proj 38031  df-bj-1upl 38038  df-bj-pr1 38041  df-bj-2upl 38051

This theorem is used by:  bj-2uplth  38061  bj-2uplex  38062