Theorem opelf 3640

Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain.

Hypothesis

Ref	Expression
opelf.1	|- D e. V

Assertion

Ref	Expression
opelf	|- ((F:A-->B /\ <.C, D>. e. F) -> (C e. A /\ D e. B))

Proof of Theorem opelf

Step	Hyp	Ref	Expression
1		fssxp 3637	. . . 4 |- (F:A-->B -> F (_ (A X. B))
2	1	sseld 2067	. . 3 |- (F:A-->B -> (<.C, D>. e. F -> <.C, D>. e. (A X. B)))
3		opelf.1	. . . 4 |- D e. V
4	3	opelxp 3214	. . 3 |- (<.C, D>. e. (A X. B) <-> (C e. A /\ D e. B))
5	2, 4	syl6ib 212	. 2 |- (F:A-->B -> (<.C, D>. e. F -> (C e. A /\ D e. B)))
6	5	imp 350	1 |- ((F:A-->B /\ <.C, D>. e. F) -> (C e. A /\ D e. B))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1811  <.cop 2411   X. cxp 3168  -->wf 3178

This theorem is referenced by:  fcoi2 3646  feu 3647  fcnvres 3648  fsn 3834

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-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-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194

Copyright terms: Public domain