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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 28277 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ) | |
2 | 1 | ancoms 455 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ) |
3 | eqid 2771 | . . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) | |
4 | 2, 3 | fmptd 6527 | . 2 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ) |
5 | brafval 29142 | . . 3 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))) | |
6 | 5 | feq1d 6170 | . 2 ⊢ (𝐴 ∈ ℋ → ((bra‘𝐴): ℋ⟶ℂ ↔ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ)) |
7 | 4, 6 | mpbird 247 | 1 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 ↦ cmpt 4863 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 ℋchil 28116 ·ih csp 28119 bracbr 28153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-hilex 28196 ax-hfi 28276 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-bra 29049 |
This theorem is referenced by: bralnfn 29147 bracl 29148 brafnmul 29150 branmfn 29304 rnbra 29306 kbass2 29316 kbass3 29317 |
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