Theorem fnopab 4548

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

Hypotheses

Ref	Expression
fnopab.1	|- (x e. A -> E!yph)
fnopab.2	|- F = {<.x, y>. | (x e. A /\ ph)}

Assertion

Ref	Expression
fnopab	|- F Fn A

Distinct variable group:   x,y,A

Proof of Theorem fnopab

Step	Hyp	Ref	Expression
1		fnopab.1	. . 3 |- (x e. A -> E!yph)
2	1	rgen 2159	. 2 |- A.x e. A E!yph
3		fnopab.2	. . 3 |- F = {<.x, y>. | (x e. A /\ ph)}
4	3	fnopabg 4546	. 2 |- (A.x e. A E!yph <-> F Fn A)
5	2, 4	mpbi 206	1 |- F Fn A

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  A.wral 2105  {copab 3395   Fn wfn 3993

This theorem is referenced by:  fnopab2 4549  fvopab3 4740

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009

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