Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funimass4f Structured version   Visualization version   GIF version

Theorem funimass4f 29890
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
Hypotheses
Ref Expression
funimass4f.1 𝑥𝐴
funimass4f.2 𝑥𝐵
funimass4f.3 𝑥𝐹
Assertion
Ref Expression
funimass4f ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Proof of Theorem funimass4f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 funimass4f.3 . . . . . 6 𝑥𝐹
21nffun 6093 . . . . 5 𝑥Fun 𝐹
3 funimass4f.1 . . . . . 6 𝑥𝐴
41nfdm 5538 . . . . . 6 𝑥dom 𝐹
53, 4nfss 3756 . . . . 5 𝑥 𝐴 ⊆ dom 𝐹
62, 5nfan 1998 . . . 4 𝑥(Fun 𝐹𝐴 ⊆ dom 𝐹)
71, 3nfima 5658 . . . . 5 𝑥(𝐹𝐴)
8 funimass4f.2 . . . . 5 𝑥𝐵
97, 8nfss 3756 . . . 4 𝑥(𝐹𝐴) ⊆ 𝐵
106, 9nfan 1998 . . 3 𝑥((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵)
11 funfvima2 6688 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
12 ssel 3757 . . . 4 ((𝐹𝐴) ⊆ 𝐵 → ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ 𝐵))
1311, 12sylan9 503 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
1410, 13ralrimi 3104 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
153, 1dfimafnf 29889 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1615adantr 472 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
178abrexss 29802 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1817adantl 473 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1916, 18eqsstrd 3801 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) ⊆ 𝐵)
2014, 19impbida 835 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  wnfc 2894  wral 3055  wrex 3056  wss 3734  dom cdm 5279  cima 5282  Fun wfun 6064  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-fv 6078
This theorem is referenced by:  ballotlem7  31048
  Copyright terms: Public domain W3C validator