MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funimass3 Structured version   Visualization version   GIF version

Theorem funimass3 6806
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 6805 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 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))

Proof of Theorem funimass3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funimass4 6712 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssel 3935 . . . . . 6 (𝐴 ⊆ dom 𝐹 → (𝑥𝐴𝑥 ∈ dom 𝐹))
3 fvimacnv 6805 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))
43ex 416 . . . . . 6 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵))))
52, 4syl9r 78 . . . . 5 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))))
65imp31 421 . . . 4 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))
76ralbidva 3186 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵)))
81, 7bitrd 282 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵)))
9 dfss3 3930 . 2 (𝐴 ⊆ (𝐹𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵))
108, 9syl6bbr 292 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  wral 3130  wss 3908  ccnv 5531  dom cdm 5532  cima 5535  Fun wfun 6328  cfv 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342
This theorem is referenced by:  funimass5  6807  funconstss  6808  fvimacnvALT  6809  fimacnv  6821  r0weon  9427  iscnp3  21847  cnpnei  21867  cnclsi  21875  cncls  21877  cncnp  21883  1stccnp  22065  txcnpi  22211  xkoco2cn  22261  xkococnlem  22262  basqtop  22314  kqnrmlem1  22346  kqnrmlem2  22347  reghmph  22396  nrmhmph  22397  elfm3  22553  rnelfm  22556  symgtgp  22709  tgpconncompeqg  22715  eltsms  22736  ucnprima  22886  plyco0  24787  plyeq0  24806  xrlimcnp  25552  rinvf1o  30383  xppreima  30402  cvmliftmolem1  32602  cvmlift2lem9  32632  cvmlift3lem6  32645  mclsppslem  32904
  Copyright terms: Public domain W3C validator