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Theorem funimass3 7054
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 7053 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 6955 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssel 3974 . . . . . 6 (𝐴 ⊆ dom 𝐹 → (𝑥𝐴𝑥 ∈ dom 𝐹))
3 fvimacnv 7053 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))
43ex 411 . . . . . 6 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵))))
52, 4syl9r 78 . . . . 5 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))))
65imp31 416 . . . 4 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵𝑥 ∈ (𝐹𝐵)))
76ralbidva 3173 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵)))
81, 7bitrd 278 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵)))
9 dfss3 3969 . 2 (𝐴 ⊆ (𝐹𝐵) ↔ ∀𝑥𝐴 𝑥 ∈ (𝐹𝐵))
108, 9bitr4di 288 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2104  wral 3059  wss 3947  ccnv 5674  dom cdm 5675  cima 5678  Fun wfun 6536  cfv 6542
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  funimass5  7055  funconstss  7056  fvimacnvALT  7057  fimacnvOLD  7071  r0weon  10009  iscnp3  22968  cnpnei  22988  cnclsi  22996  cncls  22998  cncnp  23004  1stccnp  23186  txcnpi  23332  xkoco2cn  23382  xkococnlem  23383  basqtop  23435  kqnrmlem1  23467  kqnrmlem2  23468  reghmph  23517  nrmhmph  23518  elfm3  23674  rnelfm  23677  symgtgp  23830  tgpconncompeqg  23836  eltsms  23857  ucnprima  24007  plyco0  25941  plyeq0  25960  xrlimcnp  26709  rinvf1o  32121  xppreima  32138  rhmpreimacnlem  33162  cvmliftmolem1  34570  cvmlift2lem9  34600  cvmlift3lem6  34613  mclsppslem  34872
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