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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 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
| 4 | 2, 3 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) | 
| 5 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
| 6 | fveq2 6906 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
| 7 | 4, 5, 6 | cbvmpt 5253 | . . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) | 
| 8 | 7 | eqeq2i 2750 | . 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 1540 Ⅎwnfc 2890 ↦ cmpt 5225 Fn wfn 6556 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 | 
| This theorem is referenced by: prdsgsum 19999 lgamgulm2 27079 fcomptf 32668 esumsup 34090 poimirlem16 37643 poimirlem19 37646 pwsgprod 42554 refsum2cnlem1 45042 etransclem2 46251 | 
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