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Theorem imassmpt 40402
 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 5370 . . . . 5 (𝐹𝐶) = ran (𝐹𝐶)
2 imassmpt.3 . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
32reseq1i 5640 . . . . . . 7 (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶)
4 resmpt3 5702 . . . . . . 7 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
53, 4eqtri 2802 . . . . . 6 (𝐹𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
65rneqi 5599 . . . . 5 ran (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
71, 6eqtri 2802 . . . 4 (𝐹𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
87sseq1i 3848 . . 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 2778 . . 3 (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
12 imassmpt.2 . . 3 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)
1310, 11, 12rnmptssbi 40398 . 2 (𝜑 → (ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
149, 13bitrd 271 1 (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  Ⅎwnf 1827   ∈ wcel 2107  ∀wral 3090   ∩ cin 3791   ⊆ wss 3792   ↦ cmpt 4967  ran crn 5358   ↾ cres 5359   “ cima 5360 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145 This theorem is referenced by:  limsup10exlem  40922
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