Theorem fnoprval 3042

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

Assertion

Ref	Expression
fnoprval	⊢ (F Fn (A × B) ↔ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))})

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

Proof of Theorem fnoprval

Step	Hyp	Ref	Expression
1		fnopabfv 2858	. 2 ⊢ (F Fn (A × B) ↔ F = {⟨w, z⟩∣(w ∈ (A × B) ∧ z = (F ‘w))})
2		elxp 2442	. . . . . . 7 ⊢ (w ∈ (A × B) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))
3	2	anbi1i 368	. . . . . 6 ⊢ ((w ∈ (A × B) ∧ z = (F ‘w)) ↔ (∃x∃y(w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)))
4		19.41vv 964	. . . . . . 7 ⊢ (∃x∃y((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ (∃x∃y(w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)))
5		anass 336	. . . . . . . . 9 ⊢ (((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (F ‘w))))
6		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (w = ⟨x, y⟩ → (F ‘w) = (F ‘⟨x, y⟩))
7		df-opr 3003	. . . . . . . . . . . . 13 ⊢ (xFy) = (F ‘⟨x, y⟩)
8	6, 7	syl6eqr 1142	. . . . . . . . . . . 12 ⊢ (w = ⟨x, y⟩ → (F ‘w) = (xFy))
9	8	cleq2d 1112	. . . . . . . . . . 11 ⊢ (w = ⟨x, y⟩ → (z = (F ‘w) ↔ z = (xFy)))
10	9	anbi2d 468	. . . . . . . . . 10 ⊢ (w = ⟨x, y⟩ → (((x ∈ A ∧ y ∈ B) ∧ z = (F ‘w)) ↔ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
11	10	pm5.32i 489	. . . . . . . . 9 ⊢ ((w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (F ‘w))) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
12	5, 11	bitr 151	. . . . . . . 8 ⊢ (((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
13	12	bi2ex 734	. . . . . . 7 ⊢ (∃x∃y((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
14	4, 13	bitr3 153	. . . . . 6 ⊢ ((∃x∃y(w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
15	3, 14	bitr 151	. . . . 5 ⊢ ((w ∈ (A × B) ∧ z = (F ‘w)) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
16	15	biopabi 2103	. . . 4 ⊢ {⟨w, z⟩∣(w ∈ (A × B) ∧ z = (F ‘w))} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy)))}
17		dfoprab2 3021	. . . 4 ⊢ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy)))}
18	16, 17	eqtr4 1122	. . 3 ⊢ {⟨w, z⟩∣(w ∈ (A × B) ∧ z = (F ‘w))} = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))}
19	18	cleq2i 1111	. 2 ⊢ (F = {⟨w, z⟩∣(w ∈ (A × B) ∧ z = (F ‘w))} ↔ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))})
20	1, 19	bitr 151	1 ⊢ (F Fn (A × B) ↔ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = (xFy))})

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab 2055   × cxp 2408   Fn wfn 2417   ‘cfv 2422  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  foprval 3043  mapxpen 3390

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-fv 2438  df-opr 3003  df-oprab 3004

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