Theorem fnopab2g 3602

Description: Functionality and domain of an ordered-pair class abstraction.

Hypothesis

Ref	Expression
fnopab2g.1	|- F = {<.x, y>. | (x e. A /\ y = B)}

Assertion

Ref	Expression
fnopab2g	|- (A.x e. A B e. V <-> F Fn A)

Distinct variable groups:   x,y,A   y,B

Proof of Theorem fnopab2g

Step	Hyp	Ref	Expression
1		eueq 1907	. . 3 |- (B e. V <-> E!y y = B)
2	1	ralbii 1659	. 2 |- (A.x e. A B e. V <-> A.x e. A E!y y = B)
3		fnopab2g.1	. . 3 |- F = {<.x, y>. | (x e. A /\ y = B)}
4	3	fnopabg 3601	. 2 |- (A.x e. A E!y y = B <-> F Fn A)
5	2, 4	bitr 173	1 |- (A.x e. A B e. V <-> F Fn A)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E!weu 1373  A.wral 1637  Vcvv 1802  {copab 2656   Fn wfn 3167

This theorem is referenced by:  iunfi 4543  neif 7656  grpinvf 8014  fopab2ga 10361  homcard 10426

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183

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