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Theorem funimass4f 32467
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 6571 . . . . 5 𝑥Fun 𝐹
3 funimass4f.1 . . . . . 6 𝑥𝐴
41nfdm 5947 . . . . . 6 𝑥dom 𝐹
53, 4nfss 3964 . . . . 5 𝑥 𝐴 ⊆ dom 𝐹
62, 5nfan 1894 . . . 4 𝑥(Fun 𝐹𝐴 ⊆ dom 𝐹)
71, 3nfima 6066 . . . . 5 𝑥(𝐹𝐴)
8 funimass4f.2 . . . . 5 𝑥𝐵
97, 8nfss 3964 . . . 4 𝑥(𝐹𝐴) ⊆ 𝐵
106, 9nfan 1894 . . 3 𝑥((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵)
11 funfvima2 7239 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
12 ssel 3965 . . . 4 ((𝐹𝐴) ⊆ 𝐵 → ((𝐹𝑥) ∈ (𝐹𝐴) → (𝐹𝑥) ∈ 𝐵))
1311, 12sylan9 506 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
1410, 13ralrimi 3245 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ (𝐹𝐴) ⊆ 𝐵) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
153, 1dfimafnf 32466 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1615adantr 479 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
178abrexss 32352 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1817adantl 480 . . 3 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ⊆ 𝐵)
1916, 18eqsstrd 4011 . 2 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴) ⊆ 𝐵)
2014, 19impbida 799 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {cab 2702  wnfc 2875  wral 3051  wrex 3060  wss 3939  dom cdm 5672  cima 5675  Fun wfun 6537  cfv 6543
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  ballotlem7  34212
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