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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 37016 | . 2 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
| 2 | bj-projex 36996 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1o Proj 𝐴) ∈ V) | |
| 3 | 1, 2 | eqeltrid 2845 | 1 ⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 1oc1o 8499 Proj bj-cproj 36991 pr2 bj-cpr2 37015 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-bj-proj 36992 df-bj-pr2 37016 | 
| This theorem is referenced by: bj-2uplex 37023 | 
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