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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr11val | Structured version Visualization version GIF version |
Description: Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-pr11val | ⊢ pr1 ⦅𝐴⦆ = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-1upl 36981 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
2 | bj-pr1eq 36985 | . . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴) |
4 | bj-pr1val 36987 | . 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅) | |
5 | eqid 2735 | . . 3 ⊢ ∅ = ∅ | |
6 | 5 | iftruei 4538 | . 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴 |
7 | 3, 4, 6 | 3eqtri 2767 | 1 ⊢ pr1 ⦅𝐴⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∅c0 4339 ifcif 4531 {csn 4631 × cxp 5687 tag bj-ctag 36957 ⦅bj-c1upl 36980 pr1 bj-cpr1 36983 |
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-bj-sngl 36949 df-bj-tag 36958 df-bj-proj 36974 df-bj-1upl 36981 df-bj-pr1 36984 |
This theorem is referenced by: bj-1uplth 36990 bj-1uplex 36991 bj-pr21val 36996 |
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