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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 34828 | . 2 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
2 | bj-projex 34808 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1o Proj 𝐴) ∈ V) | |
3 | 1, 2 | eqeltrid 2837 | 1 ⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3398 1oc1o 8124 Proj bj-cproj 34803 pr2 bj-cpr2 34827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-xp 5531 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-bj-proj 34804 df-bj-pr2 34828 |
This theorem is referenced by: bj-2uplex 34835 |
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