Theorem fgraphxp 41232

Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Assertion

Ref	Expression
fgraphxp	⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)})

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.

Step	Hyp	Ref	Expression
1		fgraphopab 41231	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)})
2		vex 3441	. . . . . . 7 ⊢ 𝑎 ∈ V
3		vex 3441	. . . . . . 7 ⊢ 𝑏 ∈ V
4	2, 3	op1std 7873	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑥) = 𝑎)
5	4	fveq2d 6808	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝐹‘(1st ‘𝑥)) = (𝐹‘𝑎))
6	2, 3	op2ndd 7874	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑥) = 𝑏)
7	5, 6	eqeq12d 2752	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥) ↔ (𝐹‘𝑎) = 𝑏))
8	7	rabxp 5646	. . 3 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)}
9		df-3an 1089	. . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
10	9	opabbii 5148	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
11	8, 10	eqtri 2764	. 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
12	1, 11	eqtr4di 2794	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  {crab 3303  ⟨cop 4571  {copab 5143   × cxp 5598  ⟶wf 6454  ‘cfv 6458  1^st c1st 7861  2^nd c2nd 7862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-1st 7863  df-2nd 7864

This theorem is referenced by:  hausgraph  41233