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Theorem fgraphopab 42254
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 6744 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 df-ss 3964 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
31, 2sylib 217 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4 ffn 6716 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 dffn5 6949 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
64, 5sylib 217 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
76ineq1d 4210 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
83, 7eqtr3d 2772 . 2 (𝐹:𝐴𝐵𝐹 = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
9 df-mpt 5231 . . . 4 (𝑎𝐴 ↦ (𝐹𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))}
10 df-xp 5681 . . . 4 (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}
119, 10ineq12i 4209 . . 3 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)})
12 inopab 5828 . . 3 ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))}
13 anandi 672 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)))
14 ancom 459 . . . . . . 7 ((𝑏 = (𝐹𝑎) ∧ 𝑏𝐵) ↔ (𝑏𝐵𝑏 = (𝐹𝑎)))
1514anbi2i 621 . . . . . 6 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
16 anass 467 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
17 eqcom 2737 . . . . . . 7 (𝑏 = (𝐹𝑎) ↔ (𝐹𝑎) = 𝑏)
1817anbi2i 621 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
1915, 16, 183bitr2i 298 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2013, 19bitr3i 276 . . . 4 (((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2120opabbii 5214 . . 3 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
2211, 12, 213eqtri 2762 . 2 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
238, 22eqtrdi 2786 1 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  cin 3946  wss 3947  {copab 5209  cmpt 5230   × cxp 5673   Fn wfn 6537  wf 6538  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550
This theorem is referenced by:  fgraphxp  42255
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