Theorem imassmpt 45960

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 5676	. . . 4 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
2		imassmpt.3	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	reseq1i 5976	. . . . . 6 ⊢ (𝐹 ↾ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶)
4		resmpt3 6042	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
5	3, 4	eqtri 2786	. . . . 5 ⊢ (𝐹 ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
6	5	rneqi 5929	. . . 4 ⊢ ran (𝐹 ↾ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
7	1, 6	eqtri 2786	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
8	7	sseq1i 3966	. 2 ⊢ ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷)
9		imassmpt.1	. . 3 ⊢ Ⅎ𝑥𝜑
10		eqid 2763	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
11		imassmpt.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉)
12	9, 10, 11	rnmptssbi 45958	. 2 ⊢ (𝜑 → (ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))
13	8, 12	bitrid 286	1 ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  ∀wral 3079   ∩ cin 3905   ⊆ wss 3906   ↦ cmpt 5193  ran crn 5664   ↾ cres 5665   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6540  df-fn 6541  df-f 6542

This theorem is referenced by:  limsup10exlem  46469