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Theorem imassmpt 41413
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 5561 . . . 4 (𝐹𝐶) = ran (𝐹𝐶)
2 imassmpt.3 . . . . . . 7 𝐹 = (𝑥𝐴𝐵)
32reseq1i 5842 . . . . . 6 (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶)
4 resmpt3 5899 . . . . . 6 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
53, 4eqtri 2841 . . . . 5 (𝐹𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
65rneqi 5800 . . . 4 ran (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
71, 6eqtri 2841 . . 3 (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
87sseq1i 3992 . 2 ((𝐹𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷)
9 imassmpt.1 . . 3 𝑥𝜑
10 eqid 2818 . . 3 (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
11 imassmpt.2 . . 3 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)
129, 10, 11rnmptssbi 41410 . 2 (𝜑 → (ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
138, 12syl5bb 284 1 (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  wcel 2105  wral 3135  cin 3932  wss 3933  cmpt 5137  ran crn 5549  cres 5550  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  limsup10exlem  41929
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