Theorem fgraphxp 42526

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 42525	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)})
2		vex 3472	. . . . . . 7 ⊢ 𝑎 ∈ V
3		vex 3472	. . . . . . 7 ⊢ 𝑏 ∈ V
4	2, 3	op1std 7984	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑥) = 𝑎)
5	4	fveq2d 6889	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝐹‘(1st ‘𝑥)) = (𝐹‘𝑎))
6	2, 3	op2ndd 7985	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑥) = 𝑏)
7	5, 6	eqeq12d 2742	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥) ↔ (𝐹‘𝑎) = 𝑏))
8	7	rabxp 5717	. . 3 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)}
9		df-3an 1086	. . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏))
10	9	opabbii 5208	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
11	8, 10	eqtri 2754	. 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}
12	1, 11	eqtr4di 2784	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426  ⟨cop 4629  {copab 5203   × cxp 5667  ⟶wf 6533  ‘cfv 6537  1^st c1st 7972  2^nd c2nd 7973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-1st 7974  df-2nd 7975

This theorem is referenced by:  hausgraph  42527