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Theorem fgraphopab 41952
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 6746 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 df-ss 3966 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
31, 2sylib 217 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4 ffn 6718 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 dffn5 6951 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
64, 5sylib 217 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
76ineq1d 4212 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
83, 7eqtr3d 2775 . 2 (𝐹:𝐴𝐵𝐹 = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
9 df-mpt 5233 . . . 4 (𝑎𝐴 ↦ (𝐹𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))}
10 df-xp 5683 . . . 4 (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}
119, 10ineq12i 4211 . . 3 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)})
12 inopab 5830 . . 3 ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))}
13 anandi 675 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)))
14 ancom 462 . . . . . . 7 ((𝑏 = (𝐹𝑎) ∧ 𝑏𝐵) ↔ (𝑏𝐵𝑏 = (𝐹𝑎)))
1514anbi2i 624 . . . . . 6 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
16 anass 470 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑏 = (𝐹𝑎))))
17 eqcom 2740 . . . . . . 7 (𝑏 = (𝐹𝑎) ↔ (𝐹𝑎) = 𝑏)
1817anbi2i 624 . . . . . 6 (((𝑎𝐴𝑏𝐵) ∧ 𝑏 = (𝐹𝑎)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
1915, 16, 183bitr2i 299 . . . . 5 ((𝑎𝐴 ∧ (𝑏 = (𝐹𝑎) ∧ 𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2013, 19bitr3i 277 . . . 4 (((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵)) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
2120opabbii 5216 . . 3 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
2211, 12, 213eqtri 2765 . 2 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
238, 22eqtrdi 2789 1 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cin 3948  wss 3949  {copab 5211  cmpt 5232   × cxp 5675   Fn wfn 6539  wf 6540  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by:  fgraphxp  41953
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