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Theorem fgraphxp 37615
 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 37614 . 2 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
2 vex 3201 . . . . . . 7 𝑎 ∈ V
3 vex 3201 . . . . . . 7 𝑏 ∈ V
42, 3op1std 7175 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (1st𝑥) = 𝑎)
54fveq2d 6193 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝐹‘(1st𝑥)) = (𝐹𝑎))
62, 3op2ndd 7176 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd𝑥) = 𝑏)
75, 6eqeq12d 2636 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝐹‘(1st𝑥)) = (2nd𝑥) ↔ (𝐹𝑎) = 𝑏))
87rabxp 5152 . . 3 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)}
9 df-3an 1039 . . . 4 ((𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
109opabbii 4715 . . 3 {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
118, 10eqtri 2643 . 2 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
121, 11syl6eqr 2673 1 (𝐹:𝐴𝐵𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1482   ∈ wcel 1989  {crab 2915  ⟨cop 4181  {copab 4710   × cxp 5110  ⟶wf 5882  ‘cfv 5886  1st c1st 7163  2nd c2nd 7164 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-1st 7165  df-2nd 7166 This theorem is referenced by:  hausgraph  37616
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