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Theorem fgraphxp 39689
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 39688 . 2 (𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
2 vex 3495 . . . . . . 7 𝑎 ∈ V
3 vex 3495 . . . . . . 7 𝑏 ∈ V
42, 3op1std 7688 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏⟩ → (1st𝑥) = 𝑎)
54fveq2d 6667 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝐹‘(1st𝑥)) = (𝐹𝑎))
62, 3op2ndd 7689 . . . . 5 (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd𝑥) = 𝑏)
75, 6eqeq12d 2834 . . . 4 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝐹‘(1st𝑥)) = (2nd𝑥) ↔ (𝐹𝑎) = 𝑏))
87rabxp 5593 . . 3 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)}
9 df-3an 1081 . . . 4 ((𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏) ↔ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏))
109opabbii 5124 . . 3 {⟨𝑎, 𝑏⟩ ∣ (𝑎𝐴𝑏𝐵 ∧ (𝐹𝑎) = 𝑏)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
118, 10eqtri 2841 . 2 {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)}
121, 11syl6eqr 2871 1 (𝐹:𝐴𝐵𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  cop 4563  {copab 5119   × cxp 5546  wf 6344  cfv 6348  1st c1st 7676  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-1st 7678  df-2nd 7679
This theorem is referenced by:  hausgraph  39690
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