Theorem feqmptdf 6161

Description: Deduction form of dffn5f 6162. (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 5958	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		fnrel 5903	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
4		feqmptdf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑦𝐹
6	4, 5	dfrel4 5504	. . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
7	3, 6	sylib 207	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
8		feqmptdf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
9	4, 8	nffn 5901	. . . . 5 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
10		nfv 1830	. . . . 5 ⊢ Ⅎ𝑦 𝐹 Fn 𝐴
11		fnbr 5907	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
12	11	ex 449	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
13	12	pm4.71rd 665	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
14		eqcom 2617	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
15		fnbrfvb 6146	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
16	14, 15	syl5bb 271	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
17	16	pm5.32da 671	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
18	13, 17	bitr4d 270	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))))
19	9, 10, 18	opabbid 4647	. . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
20	7, 19	eqtrd 2644	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
21		df-mpt 4645	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
22	20, 21	syl6eqr 2662	. 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 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812

This theorem is referenced by:  esumf1o  29439  volioofmpt  38887  volicofmpt  38890