Theorem fgraphopab 43164

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 6775	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		dfss2 3994	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
3	1, 2	sylib 218	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4		ffn 6747	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn5 6980	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
6	4, 5	sylib 218	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
7	6	ineq1d 4240	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
8	3, 7	eqtr3d 2782	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)))
9		df-mpt 5250	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))}
10		df-xp 5706	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)}
11	9, 10	ineq12i 4239	. . 3 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)})
12		inopab 5853	. . 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 675	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
14		ancom 460	. . . . . . 7 ⊢ ((𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵) ↔ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎)))
15	14	anbi2i 622	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎))))
16		anass 468	. . . . . 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 622	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑏 = (𝐹‘𝑎)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
19	15, 16, 18	3bitr2i 299	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
20	13, 19	bitr3i 277	. . . 4 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
21	20	opabbii 5233	. . 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 2772	. 2 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
23	8, 22	eqtrdi 2796	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  {copab 5228   ↦ cmpt 5249   × cxp 5698   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  fgraphxp  43165