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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 6727 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))) | |
2 | dffn5f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2980 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6683 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2980 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
6 | fveq2 6673 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
7 | 4, 5, 6 | cbvmpt 5170 | . . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) |
8 | 7 | eqeq2i 2837 | . 2 ⊢ (𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
9 | 1, 8 | bitri 277 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 Ⅎwnfc 2964 ↦ cmpt 5149 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 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: prdsgsum 19104 lgamgulm2 25616 fcomptf 30406 esumsup 31352 poimirlem16 34912 poimirlem19 34915 refsum2cnlem1 41300 etransclem2 42528 |
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