Theorem foprval 3043

Description: Representation of an operation abstraction in terms of its values.

Assertion

Ref	Expression
foprval	⊢ (F:(A × B)–→C ↔ (F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))} ∧ ∀x ∈ A ∀y ∈ B (xFy) ∈ C))

Distinct variable group(s):   x,y,z,A   x,B,y,z   x,C,y,z   x,F,y,z

Proof of Theorem foprval

Step	Hyp	Ref	Expression
1		ffnoprval 3041	. 2 ⊢ (F:(A × B)–→C ↔ (F Fn (A × B) ∧ ∀x ∈ A ∀y ∈ B (xFy) ∈ C))
2		fnoprval 3042	. . 3 ⊢ (F Fn (A × B) ↔ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))})
3	2	anbi1i 368	. 2 ⊢ ((F Fn (A × B) ∧ ∀x ∈ A ∀y ∈ B (xFy) ∈ C) ↔ (F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))} ∧ ∀x ∈ A ∀y ∈ B (xFy) ∈ C))
4	1, 3	bitr 151	1 ⊢ (F:(A × B)–→C ↔ (F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))} ∧ ∀x ∈ A ∀y ∈ B (xFy) ∈ C))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201   × cxp 2408   Fn wfn 2417  –→wf 2418  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  ruclem13 4897

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-oprab 3004

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