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Theorem fgraphopab 38105
Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
fgraphopab (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
Distinct variable groups:   𝐹,𝑎,𝑏   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Proof of Theorem fgraphopab
StepHypRef Expression
1 fssxp 6098 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 df-ss 3621 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
31, 2sylib 208 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4 ffn 6083 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 dffn5 6280 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
64, 5sylib 208 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
76ineq1d 3846 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
83, 7eqtr3d 2687 . 2 (𝐹:𝐴𝐵𝐹 = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
9 df-mpt 4763 . . . 4 (𝑎𝐴 ↦ (𝐹𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))}
10 df-xp 5149 . . . 4 (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}
119, 10ineq12i 3845 . . 3 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)})
12 inopab 5285 . . 3 ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))}
13 anandi 888 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)))
14 ancom 465 . . . . . . 7 ((𝑏 = (𝐹𝑎) ∧ 𝑏𝐵) ↔ (𝑏𝐵𝑏 = (𝐹𝑎)))
1514anbi2i 730 . . . . . 6 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
16 anass 682 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
17 eqcom 2658 . . . . . . 7 (𝑏 = (𝐹𝑎) ↔ (𝐹𝑎) = 𝑏)
1817anbi2i 730 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
1915, 16, 183bitr2i 288 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2013, 19bitr3i 266 . . . 4 (((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2120opabbii 4750 . . 3 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
2211, 12, 213eqtri 2677 . 2 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
238, 22syl6eq 2701 1 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cin 3606  wss 3607  {copab 4745  cmpt 4762   × cxp 5141   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  fgraphxp  38106
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