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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 34318 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-pr1eq 34309 | . . 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 34310 | . 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) | |
5 | bj-pr11val 34312 | . . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
6 | bj-pr1val 34311 | . . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅) | |
7 | 1n0 8113 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
8 | 7 | neii 3018 | . . . . . 6 ⊢ ¬ 1o = ∅ |
9 | 8 | iffalsei 4476 | . . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅ |
10 | 6, 9 | eqtri 2844 | . . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅ |
11 | 5, 10 | uneq12i 4136 | . . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅) |
12 | un0 4343 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
13 | 11, 12 | eqtri 2844 | . 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴 |
14 | 3, 4, 13 | 3eqtri 2848 | 1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3933 ∅c0 4290 ifcif 4466 {csn 4560 × cxp 5547 1oc1o 8089 tag bj-ctag 34281 ⦅bj-c1upl 34304 pr1 bj-cpr1 34307 ⦅bj-c2uple 34317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-suc 6191 df-1o 8096 df-bj-sngl 34273 df-bj-tag 34282 df-bj-proj 34298 df-bj-1upl 34305 df-bj-pr1 34308 df-bj-2upl 34318 |
This theorem is referenced by: bj-2uplth 34328 bj-2uplex 34329 |
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