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Theorem imassmpt 45269
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
imassmpt.1 𝑥𝜑
imassmpt.2 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)
imassmpt.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
imassmpt (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem imassmpt
StepHypRef Expression
1 df-ima 5698 . . . 4 (𝐹𝐶) = ran (𝐹𝐶)
2 imassmpt.3 . . . . . . 7 𝐹 = (𝑥𝐴𝐵)
32reseq1i 5993 . . . . . 6 (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶)
4 resmpt3 6056 . . . . . 6 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
53, 4eqtri 2765 . . . . 5 (𝐹𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
65rneqi 5948 . . . 4 ran (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
71, 6eqtri 2765 . . 3 (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
87sseq1i 4012 . 2 ((𝐹𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷)
9 imassmpt.1 . . 3 𝑥𝜑
10 eqid 2737 . . 3 (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
11 imassmpt.2 . . 3 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)
129, 10, 11rnmptssbi 45267 . 2 (𝜑 → (ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
138, 12bitrid 283 1 (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  wral 3061  cin 3950  wss 3951  cmpt 5225  ran crn 5686  cres 5687  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  limsup10exlem  45787
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