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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 6749 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))) | |
2 | dffn5f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2897 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6705 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
6 | fveq2 6695 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
7 | 4, 5, 6 | cbvmpt 5141 | . . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) |
8 | 7 | eqeq2i 2749 | . 2 ⊢ (𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
9 | 1, 8 | bitri 278 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 Ⅎwnfc 2877 ↦ cmpt 5120 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: prdsgsum 19320 lgamgulm2 25872 fcomptf 30669 esumsup 31723 poimirlem16 35479 poimirlem19 35482 pwsgprod 39922 refsum2cnlem1 42194 etransclem2 43395 |
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