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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr21val | Structured version Visualization version GIF version | ||
| Description: Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.) | 
| Ref | Expression | 
|---|---|
| bj-pr21val | ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-bj-2upl 37013 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
| 2 | bj-pr1eq 37004 | . . 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 37005 | . 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) | |
| 5 | bj-pr11val 37007 | . . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
| 6 | bj-pr1val 37006 | . . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅) | |
| 7 | 1n0 8527 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
| 8 | 7 | neii 2941 | . . . . . 6 ⊢ ¬ 1o = ∅ | 
| 9 | 8 | iffalsei 4534 | . . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅ | 
| 10 | 6, 9 | eqtri 2764 | . . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅ | 
| 11 | 5, 10 | uneq12i 4165 | . . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅) | 
| 12 | un0 4393 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 13 | 11, 12 | eqtri 2764 | . 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴 | 
| 14 | 3, 4, 13 | 3eqtri 2768 | 1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∪ cun 3948 ∅c0 4332 ifcif 4524 {csn 4625 × cxp 5682 1oc1o 8500 tag bj-ctag 36976 ⦅bj-c1upl 36999 pr1 bj-cpr1 37002 ⦅bj-c2uple 37012 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-suc 6389 df-1o 8507 df-bj-sngl 36968 df-bj-tag 36977 df-bj-proj 36993 df-bj-1upl 37000 df-bj-pr1 37003 df-bj-2upl 37013 | 
| This theorem is referenced by: bj-2uplth 37023 bj-2uplex 37024 | 
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