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Theorem fgraphxp 36808
 Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
fgraphxp (𝐹:𝐴𝐵𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem fgraphxp
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fgraphopab 36807 . 2 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
2 vex 3176 . . . . . . 7 𝑎 ∈ V
3 vex 3176 . . . . . . 7 𝑏 ∈ V
42, 3op1std 7069 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (1st𝑥) = 𝑎)
54fveq2d 6107 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝐹‘(1st𝑥)) = (𝐹𝑎))
62, 3op2ndd 7070 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd𝑥) = 𝑏)
75, 6eqeq12d 2625 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝐹‘(1st𝑥)) = (2nd𝑥) ↔ (𝐹𝑎) = 𝑏))
87rabxp 5078 . . 3 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)}
9 df-3an 1033 . . . 4 ((𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
109opabbii 4649 . . 3 {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
118, 10eqtri 2632 . 2 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
121, 11syl6eqr 2662 1 (𝐹:𝐴𝐵𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  ⟨cop 4131  {copab 4642   × cxp 5036  ⟶wf 5800  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  hausgraph  36809
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