Theorem fgraphopab 44163

Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Assertion

Ref	Expression
fgraphopab	⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)})

Distinct variable groups:   𝐹,𝑎,𝑏   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Proof of Theorem fgraphopab

Step	Hyp	Ref	Expression
1		fssxp 6729	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		dfss2 3917	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
3	1, 2	sylib 221	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4		ffn 6701	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn5 6935	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
6	4, 5	sylib 221	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
7	6	ineq1d 4165	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
8	3, 7	eqtr3d 2798	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
9		df-mpt 5187	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))}
10		df-xp 5657	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)}
11	9, 10	ineq12i 4164	. . 3 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)})
12		inopab 5807	. . 3 ⊢ ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)}) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))}
13		anandi 689	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
14		ancom 466	. . . . . . 7 ⊢ ((𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵) ↔ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎)))
15	14	anbi2i 635	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎))))
16		anass 474	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑏 = (𝐹‘𝑎)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎))))
17		eqcom 2768	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑎) ↔ (𝐹‘𝑎) = 𝑏)
18	17	anbi2i 635	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑏 = (𝐹‘𝑎)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
19	15, 16, 18	3bitr2i 302	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
20	13, 19	bitr3i 280	. . . 4 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
21	20	opabbii 5172	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
22	11, 12, 21	3eqtri 2788	. 2 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
23	8, 22	eqtrdi 2812	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899  {copab 5167   ↦ cmpt 5186   × cxp 5649   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539

This theorem is used by:  fgraphxp  44164