Theorem dfafn5a 47165

Description: Representation of a function in terms of its values, analogous to dffn5 6922 (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 6623	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		dfrel4v 6166	. . . 4 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
3	1, 2	sylib 218	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4		fnbr 6629	. . . . . . 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 2737	. . . . . . 7 ⊢ (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
8		fnbrafvb 47159	. . . . . . 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 579	. . . . 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 5176	. . 3 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥))})
13	3, 12	eqtrd 2765	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥))})
14		df-mpt 5192	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹'''𝑥))}
15	13, 14	eqtr4di 2783	1 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5110  {copab 5172   ↦ cmpt 5191  Rel wrel 5646   Fn wfn 6509  '''cafv 47122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522  df-aiota 47090  df-dfat 47124  df-afv 47125

This theorem is referenced by:  dfafn5b  47166  fnrnafv  47167