Theorem fgraphopab 43744

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 6715	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2		dfss2 3922	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
3	1, 2	sylib 220	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4		ffn 6687	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn5 6921	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
6	4, 5	sylib 220	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
7	6	ineq1d 4171	. . 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 5181	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))}
10		df-xp 5651	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)}
11	9, 10	ineq12i 4170	. . 3 ⊢ ((𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)})
12		inopab 5800	. . 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 686	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
14		ancom 464	. . . . . . 7 ⊢ ((𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵) ↔ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎)))
15	14	anbi2i 632	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑏 ∈ 𝐵 ∧ 𝑏 = (𝐹‘𝑎))))
16		anass 472	. . . . . 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 632	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑏 = (𝐹‘𝑎)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
19	15, 16, 18	3bitr2i 301	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ∧ (𝑏 = (𝐹‘𝑎) ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
20	13, 19	bitr3i 279	. . . 4 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 = (𝐹‘𝑎)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
21	20	opabbii 5166	. . 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)})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∩ cin 3903   ⊆ wss 3904  {copab 5161   ↦ cmpt 5180   × cxp 5643   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525

This theorem is referenced by:  fgraphxp  43745