Theorem imassmpt 41902

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 5532	. . . 4 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
2		imassmpt.3	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	reseq1i 5814	. . . . . 6 ⊢ (𝐹 ↾ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶)
4		resmpt3 5873	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
5	3, 4	eqtri 2821	. . . . 5 ⊢ (𝐹 ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
6	5	rneqi 5771	. . . 4 ⊢ ran (𝐹 ↾ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
7	1, 6	eqtri 2821	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
8	7	sseq1i 3943	. 2 ⊢ ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷)
9		imassmpt.1	. . 3 ⊢ Ⅎ𝑥𝜑
10		eqid 2798	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
11		imassmpt.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉)
12	9, 10, 11	rnmptssbi 41899	. 2 ⊢ (𝜑 → (ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))
13	8, 12	syl5bb 286	1 ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880   ⊆ wss 3881   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521   “ cima 5522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332

This theorem is referenced by:  limsup10exlem  42414