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

Theorem funimass3 7074
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 7073 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 6973 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssel 3977 . . . . . 6 (𝐴 ⊆ dom 𝐹 → (𝑥𝐴𝑥 ∈ dom 𝐹))
3 fvimacnv 7073 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))
43ex 412 . . . . . 6 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵))))
52, 4syl9r 78 . . . . 5 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))))
65imp31 417 . . . 4 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))
76ralbidva 3176 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵)))
81, 7bitrd 279 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵)))
9 dfss3 3972 . 2 (𝐴 ⊆ (𝐹𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵))
108, 9bitr4di 289 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wral 3061  wss 3951  ccnv 5684  dom cdm 5685  cima 5688  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  funimass5  7075  funconstss  7076  fvimacnvALT  7077  r0weon  10052  iscnp3  23252  cnpnei  23272  cnclsi  23280  cncls  23282  cncnp  23288  1stccnp  23470  txcnpi  23616  xkoco2cn  23666  xkococnlem  23667  basqtop  23719  kqnrmlem1  23751  kqnrmlem2  23752  reghmph  23801  nrmhmph  23802  elfm3  23958  rnelfm  23961  symgtgp  24114  tgpconncompeqg  24120  eltsms  24141  ucnprima  24291  plyco0  26231  plyeq0  26250  xrlimcnp  27011  rinvf1o  32640  xppreima  32655  rhmpreimacnlem  33883  cvmliftmolem1  35286  cvmlift2lem9  35316  cvmlift3lem6  35329  mclsppslem  35588
  Copyright terms: Public domain W3C validator