Theorem fgraphopab 44298

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 6737	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		dfss2 4094	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
3	1, 2	sylib 221	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4		ffn 6709	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn5 6943	. . . . 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 4116	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
8	3, 7	eqtr3d 2797	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
9		df-mpt 5186	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))}
10		df-xp 5656	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)}
11	9, 10	ineq12i 4115	. . 3 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)})
12		inopab 5806	. . 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 2767	. . . . . . 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 5171	. . 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 2787	. 2 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
23	8, 22	eqtrdi 2811	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 1569   ∈ wcel 2144   ⊆ wss 3895   ∩ cin 4089  {copab 5166   ↦ cmpt 5185   × cxp 5648   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546

This theorem is used by:  fgraphxp  44299