Theorem dfafn5a 46531

Description: Representation of a function in terms of its values, analogous to dffn5 6952 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
dfafn5a	⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfafn5a

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrel 6651	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		dfrel4v 6189	. . . 4 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
3	1, 2	sylib 217	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4		fnbr 6657	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
5	4	ex 412	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
6	5	pm4.71rd 562	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
7		eqcom 2735	. . . . . . 7 ⊢ (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
8		fnbrafvb 46525	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹'''𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
9	7, 8	bitrid 283	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹'''𝑥) ↔ 𝑥𝐹𝑦))
10	9	pm5.32da 578	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
11	6, 10	bitr4d 282	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥))))
12	11	opabbidv 5209	. . 3 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥))})
13	3, 12	eqtrd 2768	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥))})
14		df-mpt 5227	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥))}
15	13, 14	eqtr4di 2786	1 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   class class class wbr 5143  {copab 5205   ↦ cmpt 5226  Rel wrel 5678   Fn wfn 6538  '''cafv 46488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-res 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-aiota 46456  df-dfat 46490  df-afv 46491

This theorem is referenced by:  dfafn5b  46532  fnrnafv  46533