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Theorem funimass4f 31597
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 6525 . . . . 5 𝑥Fun 𝐹
3 funimass4f.1 . . . . . 6 𝑥𝐴
41nfdm 5907 . . . . . 6 𝑥dom 𝐹
53, 4nfss 3937 . . . . 5 𝑥 𝐴 ⊆ dom 𝐹
62, 5nfan 1903 . . . 4 𝑥(Fun 𝐹𝐴 ⊆ dom 𝐹)
71, 3nfima 6022 . . . . 5 𝑥(𝐹𝐴)
8 funimass4f.2 . . . . 5 𝑥𝐵
97, 8nfss 3937 . . . 4 𝑥(𝐹𝐴) ⊆ 𝐵
106, 9nfan 1903 . . 3 𝑥((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵)
11 funfvima2 7182 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
12 ssel 3938 . . . 4 ((𝐹𝐴) ⊆ 𝐵 → ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ 𝐵))
1311, 12sylan9 509 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
1410, 13ralrimi 3239 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
153, 1dfimafnf 31596 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1615adantr 482 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
178abrexss 31481 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1817adantl 483 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1916, 18eqsstrd 3983 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) ⊆ 𝐵)
2014, 19impbida 800 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wnfc 2884  wral 3061  wrex 3070  wss 3911  dom cdm 5634  cima 5637  Fun wfun 6491  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  ballotlem7  33192
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