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Theorem funimass4f 30297
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 6375 . . . . 5 𝑥Fun 𝐹
3 funimass4f.1 . . . . . 6 𝑥𝐴
41nfdm 5822 . . . . . 6 𝑥dom 𝐹
53, 4nfss 3964 . . . . 5 𝑥 𝐴 ⊆ dom 𝐹
62, 5nfan 1893 . . . 4 𝑥(Fun 𝐹𝐴 ⊆ dom 𝐹)
71, 3nfima 5935 . . . . 5 𝑥(𝐹𝐴)
8 funimass4f.2 . . . . 5 𝑥𝐵
97, 8nfss 3964 . . . 4 𝑥(𝐹𝐴) ⊆ 𝐵
106, 9nfan 1893 . . 3 𝑥((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵)
11 funfvima2 6988 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
12 ssel 3965 . . . 4 ((𝐹𝐴) ⊆ 𝐵 → ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ 𝐵))
1311, 12sylan9 508 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
1410, 13ralrimi 3221 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
153, 1dfimafnf 30296 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1615adantr 481 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
178abrexss 30186 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1817adantl 482 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1916, 18eqsstrd 4009 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) ⊆ 𝐵)
2014, 19impbida 797 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  {cab 2804  wnfc 2966  wral 3143  wrex 3144  wss 3940  dom cdm 5554  cima 5557  Fun wfun 6346  cfv 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360
This theorem is referenced by:  ballotlem7  31679
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