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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 36623 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-pr1eq 36614 | . . 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 36615 | . 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) | |
5 | bj-pr11val 36617 | . . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
6 | bj-pr1val 36616 | . . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅) | |
7 | 1n0 8509 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
8 | 7 | neii 2931 | . . . . . 6 ⊢ ¬ 1o = ∅ |
9 | 8 | iffalsei 4540 | . . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅ |
10 | 6, 9 | eqtri 2753 | . . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅ |
11 | 5, 10 | uneq12i 4158 | . . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅) |
12 | un0 4392 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
13 | 11, 12 | eqtri 2753 | . 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴 |
14 | 3, 4, 13 | 3eqtri 2757 | 1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3942 ∅c0 4322 ifcif 4530 {csn 4630 × cxp 5676 1oc1o 8480 tag bj-ctag 36586 ⦅bj-c1upl 36609 pr1 bj-cpr1 36612 ⦅bj-c2uple 36622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6377 df-1o 8487 df-bj-sngl 36578 df-bj-tag 36587 df-bj-proj 36603 df-bj-1upl 36610 df-bj-pr1 36613 df-bj-2upl 36623 |
This theorem is referenced by: bj-2uplth 36633 bj-2uplex 36634 |
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