Theorem dffn5f 6993

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 6980	. 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
2		dffn5f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6930	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2908	. . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥)
6		fveq2 6920	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
7	4, 5, 6	cbvmpt 5277	. . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
8	7	eqeq2i 2753	. 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 1537  Ⅎwnfc 2893   ↦ cmpt 5249   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  prdsgsum  20023  lgamgulm2  27097  fcomptf  32676  esumsup  34053  poimirlem16  37596  poimirlem19  37599  pwsgprod  42499  refsum2cnlem1  44937  etransclem2  46157