Theorem resopab2 3398

Description: Restriction of a class abstraction of ordered pairs.

Assertion

Ref	Expression
resopab2	|- (A (_ B -> ({<.x, y>. | (x e. B /\ ph)} |` A) = {<.x, y>. | (x e. A /\ ph)})

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

Proof of Theorem resopab2

Step	Hyp	Ref	Expression
1		ssel 2063	. . . . . 6 |- (A (_ B -> (x e. A -> x e. B))
2		pm4.71 635	. . . . . 6 |- ((x e. A -> x e. B) <-> (x e. A <-> (x e. A /\ x e. B)))
3	1, 2	sylib 198	. . . . 5 |- (A (_ B -> (x e. A <-> (x e. A /\ x e. B)))
4	3	anbi1d 617	. . . 4 |- (A (_ B -> ((x e. A /\ ph) <-> ((x e. A /\ x e. B) /\ ph)))
5		anass 439	. . . 4 |- (((x e. A /\ x e. B) /\ ph) <-> (x e. A /\ (x e. B /\ ph)))
6	4, 5	syl6rbb 537	. . 3 |- (A (_ B -> ((x e. A /\ (x e. B /\ ph)) <-> (x e. A /\ ph)))
7	6	opabbidv 2670	. 2 |- (A (_ B -> {<.x, y>. | (x e. A /\ (x e. B /\ ph))} = {<.x, y>. | (x e. A /\ ph)})
8		resopab 3395	. 2 |- ({<.x, y>. | (x e. B /\ ph)} |` A) = {<.x, y>. | (x e. A /\ (x e. B /\ ph))}
9	7, 8	syl5eq 1519	1 |- (A (_ B -> ({<.x, y>. | (x e. B /\ ph)} |` A) = {<.x, y>. | (x e. A /\ ph)})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   (_ wss 2047  {copab 2666   |` cres 3172

This theorem is referenced by:  geolim1i 7238  efseq0ex 7311  reeff1 7410  ipasslem7 8496

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185  df-res 3190

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