Theorem fgraphopab 43655

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 6689	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		dfss2 3908	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
3	1, 2	sylib 219	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4		ffn 6662	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn5 6892	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
6	4, 5	sylib 219	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
7	6	ineq1d 4155	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
8	3, 7	eqtr3d 2777	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
9		df-mpt 5161	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))}
10		df-xp 5631	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)}
11	9, 10	ineq12i 4154	. . 3 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)})
12		inopab 5779	. . 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 682	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
14		ancom 461	. . . . . . 7 ⊢ ((𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵) ↔ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎)))
15	14	anbi2i 629	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎))))
16		anass 469	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑏 = (𝐹‘𝑎)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎))))
17		eqcom 2747	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑎) ↔ (𝐹‘𝑎) = 𝑏)
18	17	anbi2i 629	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑏 = (𝐹‘𝑎)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
19	15, 16, 18	3bitr2i 300	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
20	13, 19	bitr3i 278	. . . 4 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
21	20	opabbii 5146	. . 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 2767	. 2 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
23	8, 22	eqtrdi 2791	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3889   ⊆ wss 3890  {copab 5141   ↦ cmpt 5160   × cxp 5623   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  fgraphxp  43656