Theorem fnoprab2 3039

Description: Functionality and domain of an operation abstraction.

Hypotheses

Ref	Expression
fnoprab2.1	|- C e. V
fnoprab2.2	|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}

Assertion

Ref	Expression
fnoprab2	|- F Fn (A X. 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 e. V
2	1	eueq1 1428	. . . 4 |- E!z z = C
3	2	a1i 7	. . 3 |- ((x e. A /\ y e. B) -> E!z z = C)
4	3	fnoprab 3038	. 2 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn {<.x, y>. | (x e. A /\ y e. B)}
5		fnoprab2.2	. . . 4 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
6		fneq1 2718	. . . 4 |- (F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} -> (F Fn (A X. B) <-> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn (A X. B)))
7	5, 6	ax-mp 6	. . 3 |- (F Fn (A X. B) <-> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn (A X. B))
8		df-xp 2424	. . . 4 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
9		fneq2 2719	. . . 4 |- ((A X. B) = {<.x, y>. | (x e. A /\ y e. B)} -> ({<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn (A X. B) <-> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn {<.x, y>. | (x e. A /\ y e. B)}))
10	8, 9	ax-mp 6	. . 3 |- ({<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn (A X. B) <-> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn {<.x, y>. | (x e. A /\ y e. B)})
11	7, 10	bitr 151	. 2 |- (F Fn (A X. B) <-> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} Fn {<.x, y>. | (x e. A /\ y e. B)})
12	4, 11	mpbir 165	1 |- F Fn (A X. B)

Syntax hints:   <-> wb 127   /\ wa 196  E!weu 1007   = wceq 1091   e. wcel 1092  Vcvv 1348  {copab 2055   X. cxp 2408   Fn wfn 2417  {copab2 3002

This theorem is referenced by:  dmoprab2 3040  elrnoprab 3054  df1st2 3098  fnoa 3117  fnom 3118  fnmap 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

metamath.org