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Theorem fgraphxp 39804
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 39803 . 2 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
2 vex 3498 . . . . . . 7 𝑎 ∈ V
3 vex 3498 . . . . . . 7 𝑏 ∈ V
42, 3op1std 7693 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (1st𝑥) = 𝑎)
54fveq2d 6669 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝐹‘(1st𝑥)) = (𝐹𝑎))
62, 3op2ndd 7694 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd𝑥) = 𝑏)
75, 6eqeq12d 2837 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝐹‘(1st𝑥)) = (2nd𝑥) ↔ (𝐹𝑎) = 𝑏))
87rabxp 5595 . . 3 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)}
9 df-3an 1085 . . . 4 ((𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
109opabbii 5126 . . 3 {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
118, 10eqtri 2844 . 2 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
121, 11syl6eqr 2874 1 (𝐹:𝐴𝐵𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  {crab 3142  cop 4567  {copab 5121   × cxp 5548  wf 6346  cfv 6350  1st c1st 7681  2nd c2nd 7682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-1st 7683  df-2nd 7684
This theorem is referenced by:  hausgraph  39805
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