Theorem fnoprab2 3039

Description: Functionality and domain of an operation abstraction.

Hypotheses

Ref	Expression
fnoprab2.1	⊢ C ∈ V
fnoprab2.2	⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)}

Assertion

Ref	Expression
fnoprab2	⊢ F Fn (A × B)

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

Proof of Theorem fnoprab2

Step	Hyp	Ref	Expression
1		fnoprab2.1	. . . . 5 ⊢ C ∈ V
2	1	eueq1 1428	. . . 4 ⊢ ∃!z z = C
3	2	a1i 7	. . 3 ⊢ ((x ∈ A ∧ y ∈ B) → ∃!z z = C)
4	3	fnoprab 3038	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn {⟨x, y⟩∣(x ∈ A ∧ y ∈ B)}
5		fnoprab2.2	. . . 4 ⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)}
6		fneq1 2718	. . . 4 ⊢ (F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} → (F Fn (A × B) ↔ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn (A × B)))
7	5, 6	ax-mp 6	. . 3 ⊢ (F Fn (A × B) ↔ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn (A × B))
8		df-xp 2424	. . . 4 ⊢ (A × B) = {⟨x, y⟩∣(x ∈ A ∧ y ∈ B)}
9		fneq2 2719	. . . 4 ⊢ ((A × B) = {⟨x, y⟩∣(x ∈ A ∧ y ∈ B)} → ({⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn (A × B) ↔ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn {⟨x, y⟩∣(x ∈ A ∧ y ∈ B)}))
10	8, 9	ax-mp 6	. . 3 ⊢ ({⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn (A × B) ↔ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn {⟨x, y⟩∣(x ∈ A ∧ y ∈ B)})
11	7, 10	bitr 151	. 2 ⊢ (F Fn (A × B) ↔ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} Fn {⟨x, y⟩∣(x ∈ A ∧ y ∈ B)})
12	4, 11	mpbir 165	1 ⊢ F Fn (A × B)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {copab 2055   × cxp 2408   Fn wfn 2417  {copab2 3002

This theorem is referenced by:  dmoprab2 3040  elrnoprab 3054  df1st2 3098  fnoa 3117  fnom 3118   3262  mapxpen 3390  unxpdomlem 3649  qnnen 4931

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-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-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-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432  df-fn 2433  df-oprab 3004

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