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Theorem fgraphopab 41566
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 6701 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 df-ss 3932 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) ↔ (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
31, 2sylib 217 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = 𝐹)
4 ffn 6673 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 dffn5 6906 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
64, 5sylib 217 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
76ineq1d 4176 . . 3 (𝐹:𝐴𝐵 → (𝐹 ∩ (𝐴 × 𝐵)) = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
83, 7eqtr3d 2779 . 2 (𝐹:𝐴𝐵𝐹 = ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)))
9 df-mpt 5194 . . . 4 (𝑎𝐴 ↦ (𝐹𝑎)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))}
10 df-xp 5644 . . . 4 (𝐴 × 𝐵) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)}
119, 10ineq12i 4175 . . 3 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = ({⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏 = (𝐹𝑎))} ∩ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵)})
12 inopab 5790 . . 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 2744 . . . . . . 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 5177 . . 3 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏 = (𝐹𝑎)) ∧ (𝑎𝐴𝑏𝐵))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
2211, 12, 213eqtri 2769 . 2 ((𝑎𝐴 ↦ (𝐹𝑎)) ∩ (𝐴 × 𝐵)) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
238, 22eqtrdi 2793 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 3914  wss 3915  {copab 5172  cmpt 5193   × cxp 5636   Fn wfn 6496  wf 6497  cfv 6501
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509
This theorem is referenced by:  fgraphxp  41567
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