Theorem funimass3 6013

Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 6012 would be the special case of A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

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

Proof of Theorem funimass3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimass4 5930	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
2		ssel 3412	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		fvimacnv 6012	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
4	3	ex 441	. . . . . 6 |- ( Fun F -> ( x e. dom F -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) )
5	2, 4	syl9r 73	. . . . 5 |- ( Fun F -> ( A C_ dom F -> ( x e. A -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) ) )
6	5	imp31 439	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
7	6	ralbidva 2828	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( A. x e. A ( F ` x ) e. B <-> A. x e. A x e. ( `' F " B ) ) )
8	1, 7	bitrd 261	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A x e. ( `' F " B ) ) )
9		dfss3 3408	. 2 |- ( A C_ ( `' F " B ) <-> A. x e. A x e. ( `' F " B ) )
10	8, 9	syl6bbr 271	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   e. wcel 1904   A.wral 2756   C_ wss 3390   `'ccnv 4838   dom cdm 4839   "cima 4842   Fun wfun 5583   ` cfv 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597

This theorem is referenced by:  funimass5  6014  funconstss  6015  fvimacnvALT  6016  fimacnv  6027  r0weon  8461  iscnp3  20337  cnpnei  20357  cnclsi  20365  cncls  20367  cncnp  20373  1stccnp  20554  txcnpi  20700  xkoco2cn  20750  xkococnlem  20751  basqtop  20803  kqnrmlem1  20835  kqnrmlem2  20836  reghmph  20885  nrmhmph  20886  elfm3  21043  rnelfm  21046  symgtgp  21194  tgpconcompeqg  21204  eltsms  21225  ucnprima  21375  plyco0  23225  plyeq0  23244  xrlimcnp  23973  rinvf1o  28306  xppreima  28324  cvmliftmolem1  30076  cvmlift2lem9  30106  cvmlift3lem6  30119  mclsppslem  30293