Theorem fnopab 3613

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 1696	. 2 |- A.x e. A E!yph
3		fnopab.2	. . 3 |- F = {<.x, y>. | (x e. A /\ ph)}
4	3	fnopabg 3611	. 2 |- (A.x e. A E!yph <-> F Fn A)
5	2, 4	mpbi 189	1 |- F Fn A

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  E!weu 1379  A.wral 1643  {copab 2662   Fn wfn 3173

This theorem is referenced by:  fnopab2 3614  fvopab3 3772

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fn 3189

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