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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr1ex | Structured version Visualization version GIF version |
Description: Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-pr1ex | ⊢ (𝐴 ∈ 𝑉 → pr1 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-pr1 33937 | . 2 ⊢ pr1 𝐴 = (∅ Proj 𝐴) | |
2 | bj-projex 33931 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ Proj 𝐴) ∈ V) | |
3 | 1, 2 | syl5eqel 2887 | 1 ⊢ (𝐴 ∈ 𝑉 → pr1 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 Vcvv 3437 ∅c0 4211 Proj bj-cproj 33926 pr1 bj-cpr1 33936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-xp 5449 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-bj-proj 33927 df-bj-pr1 33937 |
This theorem is referenced by: bj-1uplex 33944 bj-2uplex 33958 |
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