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Mirrors > Home > MPE Home > Th. List > dffn5f | Structured version Visualization version GIF version |
Description: Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
dffn5f.1 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
dffn5f | ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5 6967 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))) | |
2 | dffn5f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6917 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
6 | fveq2 6907 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
7 | 4, 5, 6 | cbvmpt 5259 | . . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) |
8 | 7 | eqeq2i 2748 | . 2 ⊢ (𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
9 | 1, 8 | bitri 275 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 Ⅎwnfc 2888 ↦ cmpt 5231 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: prdsgsum 20014 lgamgulm2 27094 fcomptf 32675 esumsup 34070 poimirlem16 37623 poimirlem19 37626 pwsgprod 42531 refsum2cnlem1 44975 etransclem2 46192 |
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