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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 36994 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-pr1eq 36985 | . . 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 36986 | . 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) | |
5 | bj-pr11val 36988 | . . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
6 | bj-pr1val 36987 | . . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅) | |
7 | 1n0 8525 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
8 | 7 | neii 2940 | . . . . . 6 ⊢ ¬ 1o = ∅ |
9 | 8 | iffalsei 4541 | . . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅ |
10 | 6, 9 | eqtri 2763 | . . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅ |
11 | 5, 10 | uneq12i 4176 | . . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅) |
12 | un0 4400 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
13 | 11, 12 | eqtri 2763 | . 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴 |
14 | 3, 4, 13 | 3eqtri 2767 | 1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3961 ∅c0 4339 ifcif 4531 {csn 4631 × cxp 5687 1oc1o 8498 tag bj-ctag 36957 ⦅bj-c1upl 36980 pr1 bj-cpr1 36983 ⦅bj-c2uple 36993 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-1o 8505 df-bj-sngl 36949 df-bj-tag 36958 df-bj-proj 36974 df-bj-1upl 36981 df-bj-pr1 36984 df-bj-2upl 36994 |
This theorem is referenced by: bj-2uplth 37004 bj-2uplex 37005 |
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