Theorem dffn5f 6899

Description: Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)

Hypothesis

Ref	Expression
dffn5f.1	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
dffn5f	⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem dffn5f

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 6886	. 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
2		dffn5f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6838	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2894	. . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥)
6		fveq2 6828	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
7	4, 5, 6	cbvmpt 5195	. . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
8	7	eqeq2i 2744	. 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)))

Syntax hints:   ↔ wb 206   = wceq 1541  Ⅎwnfc 2879   ↦ cmpt 5174   Fn wfn 6482  ‘cfv 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6443  df-fun 6489  df-fn 6490  df-fv 6495

This theorem is referenced by:  prdsgsum  19899  lgamgulm2  26979  fcomptf  32647  esumsup  34109  poimirlem16  37682  poimirlem19  37685  pwsgprod  42643  refsum2cnlem1  45139  etransclem2  46339