Theorem imassmpt 45234

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	⊢ Ⅎ𝑥𝜑
imassmpt.2	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉)
imassmpt.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
imassmpt	⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem imassmpt

Step	Hyp	Ref	Expression
1		df-ima 5667	. . . 4 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
2		imassmpt.3	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	reseq1i 5962	. . . . . 6 ⊢ (𝐹 ↾ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶)
4		resmpt3 6025	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
5	3, 4	eqtri 2758	. . . . 5 ⊢ (𝐹 ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
6	5	rneqi 5917	. . . 4 ⊢ ran (𝐹 ↾ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
7	1, 6	eqtri 2758	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
8	7	sseq1i 3987	. 2 ⊢ ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷)
9		imassmpt.1	. . 3 ⊢ Ⅎ𝑥𝜑
10		eqid 2735	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
11		imassmpt.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉)
12	9, 10, 11	rnmptssbi 45232	. 2 ⊢ (𝜑 → (ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))
13	8, 12	bitrid 283	1 ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  ∀wral 3051   ∩ cin 3925   ⊆ wss 3926   ↦ cmpt 5201  ran crn 5655   ↾ cres 5656   “ cima 5657

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6532  df-fn 6533  df-f 6534

This theorem is referenced by:  limsup10exlem  45749