Theorem funimass3 5979

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 5978 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 5899	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
2		ssel 3483	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		fvimacnv 5978	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
4	3	ex 432	. . . . . 6 |- ( Fun F -> ( x e. dom F -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) )
5	2, 4	syl9r 72	. . . . 5 |- ( Fun F -> ( A C_ dom F -> ( x e. A -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) ) )
6	5	imp31 430	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
7	6	ralbidva 2890	. . 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 253	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A x e. ( `' F " B ) ) )
9		dfss3 3479	. 2 |- ( A C_ ( `' F " B ) <-> A. x e. A x e. ( `' F " B ) )
10	8, 9	syl6bbr 263	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   e. wcel 1823   A.wral 2804   C_ wss 3461   `'ccnv 4987   dom cdm 4988   "cima 4991   Fun wfun 5564   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578

This theorem is referenced by:  funimass5  5980  funconstss  5981  fvimacnvALT  5982  fimacnv  5995  r0weon  8381  iscnp3  19912  cnpnei  19932  cnclsi  19940  cncls  19942  cncnp  19948  1stccnp  20129  txcnpi  20275  xkoco2cn  20325  xkococnlem  20326  basqtop  20378  kqnrmlem1  20410  kqnrmlem2  20411  reghmph  20460  nrmhmph  20461  elfm3  20617  rnelfm  20620  symgtgp  20766  tgpconcompeqg  20776  eltsms  20797  ucnprima  20951  plyco0  22755  plyeq0  22774  xrlimcnp  23496  rinvf1o  27691  xppreima  27708  cvmliftmolem1  28990  cvmlift2lem9  29020  cvmlift3lem6  29033  mclsppslem  29207