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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 34895 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-pr1eq 34886 | . . 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 34887 | . 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) | |
5 | bj-pr11val 34889 | . . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
6 | bj-pr1val 34888 | . . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅) | |
7 | 1n0 8210 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
8 | 7 | neii 2937 | . . . . . 6 ⊢ ¬ 1o = ∅ |
9 | 8 | iffalsei 4439 | . . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅ |
10 | 6, 9 | eqtri 2762 | . . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅ |
11 | 5, 10 | uneq12i 4065 | . . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅) |
12 | un0 4295 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
13 | 11, 12 | eqtri 2762 | . 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴 |
14 | 3, 4, 13 | 3eqtri 2766 | 1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∪ cun 3855 ∅c0 4227 ifcif 4429 {csn 4531 × cxp 5538 1oc1o 8184 tag bj-ctag 34858 ⦅bj-c1upl 34881 pr1 bj-cpr1 34884 ⦅bj-c2uple 34894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-suc 6208 df-1o 8191 df-bj-sngl 34850 df-bj-tag 34859 df-bj-proj 34875 df-bj-1upl 34882 df-bj-pr1 34885 df-bj-2upl 34895 |
This theorem is referenced by: bj-2uplth 34905 bj-2uplex 34906 |
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