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Theorem fgraphopab 36700
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 5858 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 df-ss 3458 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
31, 2sylib 206 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4 ffn 5843 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 dffn5 6034 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
64, 5sylib 206 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
76ineq1d 3678 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
83, 7eqtr3d 2550 . 2 (𝐹:𝐴𝐵𝐹 = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
9 df-mpt 4543 . . . 4 (𝑎𝐴 ↦ (𝐹𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))}
10 df-xp 4938 . . . 4 (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}
119, 10ineq12i 3677 . . 3 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)})
12 inopab 5066 . . 3 ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))}
13 anandi 866 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)))
14 ancom 464 . . . . . . 7 ((𝑏 = (𝐹𝑎) ∧ 𝑏𝐵) ↔ (𝑏𝐵𝑏 = (𝐹𝑎)))
1514anbi2i 725 . . . . . 6 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
16 anass 678 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
17 eqcom 2521 . . . . . . 7 (𝑏 = (𝐹𝑎) ↔ (𝐹𝑎) = 𝑏)
1817anbi2i 725 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
1915, 16, 183bitr2i 286 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2013, 19bitr3i 264 . . . 4 (((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2120opabbii 4547 . . 3 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
2211, 12, 213eqtri 2540 . 2 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
238, 22syl6eq 2564 1 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  cin 3443  wss 3444  {copab 4540  cmpt 4541   × cxp 4930   Fn wfn 5684  wf 5685  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-fv 5697
This theorem is referenced by:  fgraphxp  36701
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