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Theorem imassmpt 39795
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 5156 . . . . 5 (𝐹𝐶) = ran (𝐹𝐶)
2 imassmpt.3 . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
32reseq1i 5424 . . . . . . 7 (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶)
4 resmpt3 5485 . . . . . . 7 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
53, 4eqtri 2673 . . . . . 6 (𝐹𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
65rneqi 5384 . . . . 5 ran (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
71, 6eqtri 2673 . . . 4 (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
87sseq1i 3662 . . 3 ((𝐹𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷)
98a1i 11 . 2 (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷))
10 imassmpt.1 . . 3 𝑥𝜑
11 eqid 2651 . . 3 (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
12 imassmpt.2 . . 3 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)
1310, 11, 12rnmptssbi 39791 . 2 (𝜑 → (ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
149, 13bitrd 268 1 (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wral 2941  cin 3606  wss 3607  cmpt 4762  ran crn 5144  cres 5145  cima 5146
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-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-mpt 4763  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-f 5930  df-fv 5934
This theorem is referenced by:  limsup10exlem  40322
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