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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr2ex | Structured version Visualization version GIF version |
Description: Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-pr2ex | ⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-pr2 35834 | . 2 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
2 | bj-projex 35814 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1o Proj 𝐴) ∈ V) | |
3 | 1, 2 | eqeltrid 2838 | 1 ⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3475 1oc1o 8454 Proj bj-cproj 35809 pr2 bj-cpr2 35833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-bj-proj 35810 df-bj-pr2 35834 |
This theorem is referenced by: bj-2uplex 35841 |
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