Theorem dffn5f 6822

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 6810	. 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
2		dffn5f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6766	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2906	. . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥)
6		fveq2 6756	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
7	4, 5, 6	cbvmpt 5181	. . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
8	7	eqeq2i 2751	. 2 ⊢ (𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	1, 8	bitri 274	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Syntax hints:   ↔ wb 205   = wceq 1539  Ⅎwnfc 2886   ↦ cmpt 5153   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  prdsgsum  19497  lgamgulm2  26090  fcomptf  30897  esumsup  31957  poimirlem16  35720  poimirlem19  35723  pwsgprod  40194  refsum2cnlem1  42469  etransclem2  43667