Theorem opelf 3631

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 3628	. . . 4 |- (F:A-->B -> F (_ (A X. B))
2	1	sseld 2063	. . 3 |- (F:A-->B -> (<.C, D>. e. F -> <.C, D>. e. (A X. B)))
3		opelf.1	. . . 4 |- D e. V
4	3	opelxp 3209	. . 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 956  Vcvv 1807  <.cop 2407   X. cxp 3163  -->wf 3173

This theorem is referenced by:  fcoi2 3637  feu 3638  fcnvres 3639  fsn 3825

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189

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