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Theorem imassmpt 41535
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 5567 . . . 4 (𝐹𝐶) = ran (𝐹𝐶)
2 imassmpt.3 . . . . . . 7 𝐹 = (𝑥𝐴𝐵)
32reseq1i 5848 . . . . . 6 (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶)
4 resmpt3 5905 . . . . . 6 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
53, 4eqtri 2844 . . . . 5 (𝐹𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
65rneqi 5806 . . . 4 ran (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
71, 6eqtri 2844 . . 3 (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
87sseq1i 3994 . 2 ((𝐹𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷)
9 imassmpt.1 . . 3 𝑥𝜑
10 eqid 2821 . . 3 (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
11 imassmpt.2 . . 3 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)
129, 10, 11rnmptssbi 41532 . 2 (𝜑 → (ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
138, 12syl5bb 285 1 (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wnf 1780  wcel 2110  wral 3138  cin 3934  wss 3935  cmpt 5145  ran crn 5555  cres 5556  cima 5557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  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 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362
This theorem is referenced by:  limsup10exlem  42051
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