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Theorem funimass4f 29565
 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 5949 . . . . 5 𝑥Fun 𝐹
3 funimass4f.1 . . . . . 6 𝑥𝐴
41nfdm 5399 . . . . . 6 𝑥dom 𝐹
53, 4nfss 3629 . . . . 5 𝑥 𝐴 ⊆ dom 𝐹
62, 5nfan 1868 . . . 4 𝑥(Fun 𝐹𝐴 ⊆ dom 𝐹)
71, 3nfima 5509 . . . . 5 𝑥(𝐹𝐴)
8 funimass4f.2 . . . . 5 𝑥𝐵
97, 8nfss 3629 . . . 4 𝑥(𝐹𝐴) ⊆ 𝐵
106, 9nfan 1868 . . 3 𝑥((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵)
11 funfvima2 6533 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
12 ssel 3630 . . . 4 ((𝐹𝐴) ⊆ 𝐵 → ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ 𝐵))
1311, 12sylan9 690 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
1410, 13ralrimi 2986 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
153, 1dfimafnf 29564 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1615adantr 480 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
178abrexss 29476 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1817adantl 481 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1916, 18eqsstrd 3672 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) ⊆ 𝐵)
2014, 19impbida 895 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  Ⅎwnfc 2780  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607  dom cdm 5143   “ cima 5146  Fun wfun 5920  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934 This theorem is referenced by:  ballotlem7  30725
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