Theorem funimass3 3806

Description: A kind of contraposition law that infers an image subclass from a subclass of a pre-image. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "Likely this could be proved directly, and fvimacnv 3805 would be the special case of A being a singleton, but it works this way round too.")

Assertion

Ref	Expression
funimass3	|- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))

Proof of Theorem funimass3

Step	Hyp	Ref	Expression
1		funimass4 3763	. . 3 |- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A.x e. A (F` x) e. B))
2		ssel 2063	. . . . . 6 |- (A (_ dom F -> (x e. A -> x e. dom F))
3		fvimacnv 3805	. . . . . . 7 |- ((Fun F /\ x e. dom F) -> ((F` x) e. B <-> x e. (`'F"B)))
4	3	ex 373	. . . . . 6 |- (Fun F -> (x e. dom F -> ((F` x) e. B <-> x e. (`'F"B))))
5	2, 4	syl9r 58	. . . . 5 |- (Fun F -> (A (_ dom F -> (x e. A -> ((F` x) e. B <-> x e. (`'F"B)))))
6	5	imp31 362	. . . 4 |- (((Fun F /\ A (_ dom F) /\ x e. A) -> ((F` x) e. B <-> x e. (`'F"B)))
7	6	ralbidva 1659	. . 3 |- ((Fun F /\ A (_ dom F) -> (A.x e. A (F` x) e. B <-> A.x e. A x e. (`'F"B)))
8	1, 7	bitrd 528	. 2 |- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A.x e. A x e. (`'F"B)))
9		dfss3 2059	. 2 |- (A (_ (`'F"B) <-> A.x e. A x e. (`'F"B))
10	8, 9	syl6bbr 538	1 |- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958  A.wral 1645   (_ wss 2047  `'ccnv 3169  dom cdm 3170  "cima 3173  Fun wfun 3176  ` cfv 3182

This theorem is referenced by:  funimass5 3807  funconstss 3808  fvimacnvALT 3809  fimacnv 3810  iscnp2 7761  cncnplem4 7777  metcnp 7887  metcnp3 7896

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  ax-un 2866

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-ral 1649  df-rex 1650  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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