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Mirrors > Home > HSE Home > Th. List > brafn | Structured version Visualization version GIF version |
Description: The bra function is a functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
brafn | ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hicl 27321 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ) | |
2 | 1 | ancoms 468 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ) |
3 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) | |
4 | 2, 3 | fmptd 6292 | . 2 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ) |
5 | brafval 28186 | . . 3 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))) | |
6 | 5 | feq1d 5943 | . 2 ⊢ (𝐴 ∈ ℋ → ((bra‘𝐴): ℋ⟶ℂ ↔ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ)) |
7 | 4, 6 | mpbird 246 | 1 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℋchil 27160 ·ih csp 27163 bracbr 27197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-hilex 27240 ax-hfi 27320 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-bra 28093 |
This theorem is referenced by: bralnfn 28191 bracl 28192 brafnmul 28194 branmfn 28348 rnbra 28350 kbass2 28360 kbass3 28361 |
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