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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 6718 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))) | |
2 | dffn5f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6674 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
6 | fveq2 6664 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
7 | 4, 5, 6 | cbvmpt 5159 | . . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) |
8 | 7 | eqeq2i 2834 | . 2 ⊢ (𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
9 | 1, 8 | bitri 276 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 Ⅎwnfc 2961 ↦ cmpt 5138 Fn wfn 6344 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 |
This theorem is referenced by: prdsgsum 19032 lgamgulm2 25541 fcomptf 30332 esumsup 31248 poimirlem16 34790 poimirlem19 34793 refsum2cnlem1 41174 etransclem2 42402 |
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