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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 32192 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) | |
2 | bj-pr1eq 32183 | . . 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 32184 | . 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1𝑜} × tag 𝐵)) | |
5 | bj-pr11val 32186 | . . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
6 | bj-pr1val 32185 | . . . . 5 ⊢ pr1 ({1𝑜} × tag 𝐵) = if(1𝑜 = ∅, 𝐵, ∅) | |
7 | 1n0 7462 | . . . . . . 7 ⊢ 1𝑜 ≠ ∅ | |
8 | 7 | neii 2784 | . . . . . 6 ⊢ ¬ 1𝑜 = ∅ |
9 | 8 | iffalsei 4046 | . . . . 5 ⊢ if(1𝑜 = ∅, 𝐵, ∅) = ∅ |
10 | 6, 9 | eqtri 2632 | . . . 4 ⊢ pr1 ({1𝑜} × tag 𝐵) = ∅ |
11 | 5, 10 | uneq12i 3727 | . . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1𝑜} × tag 𝐵)) = (𝐴 ∪ ∅) |
12 | un0 3919 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
13 | 11, 12 | eqtri 2632 | . 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1𝑜} × tag 𝐵)) = 𝐴 |
14 | 3, 4, 13 | 3eqtri 2636 | 1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 ∅c0 3874 ifcif 4036 {csn 4125 × cxp 5036 1𝑜c1o 7440 tag bj-ctag 32155 ⦅bj-c1upl 32178 pr1 bj-cpr1 32181 ⦅bj-c2uple 32191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-1o 7447 df-bj-sngl 32147 df-bj-tag 32156 df-bj-proj 32172 df-bj-1upl 32179 df-bj-pr1 32182 df-bj-2upl 32192 |
This theorem is referenced by: bj-2uplth 32202 bj-2uplex 32203 |
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