Theorem feqmptdf 6904

Description: Deduction form of dffn5f 6905. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)

Hypotheses

Ref	Expression
feqmptdf.1	⊢ Ⅎ𝑥𝐴
feqmptdf.2	⊢ Ⅎ𝑥𝐹
feqmptdf.3	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
feqmptdf	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Proof of Theorem feqmptdf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		feqmptdf.3	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		ffn 6662	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		fnrel 6594	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
4		feqmptdf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑦𝐹
6	4, 5	dfrel4 6149	. . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
7	3, 6	sylib 218	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
8		feqmptdf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
9	4, 8	nffn 6591	. . . . 5 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
10		nfv 1915	. . . . 5 ⊢ Ⅎ𝑦 𝐹 Fn 𝐴
11		fnbr 6600	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
12	11	ex 412	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
13	12	pm4.71rd 562	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
14		eqcom 2743	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
15		fnbrfvb 6884	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
16	14, 15	bitrid 283	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
17	16	pm5.32da 579	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
18	13, 17	bitr4d 282	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))))
19	9, 10, 18	opabbid 5163	. . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
20	7, 19	eqtrd 2771	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
21		df-mpt 5180	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
22	20, 21	eqtr4di 2789	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
23	1, 2, 22	3syl 18	1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2883   class class class wbr 5098  {copab 5160   ↦ cmpt 5179  Rel wrel 5629   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  esumf1o  34207  feqresmptf  45475  liminfvaluz3  46040  liminfvaluz4  46043  volioofmpt  46238  volicofmpt  46241