Theorem imassmpt 46243

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 5664	. . . 4 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
2		imassmpt.3	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	reseq1i 5966	. . . . . 6 ⊢ (𝐹 ↾ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶)
4		resmpt3 6030	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
5	3, 4	eqtri 2784	. . . . 5 ⊢ (𝐹 ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
6	5	rneqi 5919	. . . 4 ⊢ ran (𝐹 ↾ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
7	1, 6	eqtri 2784	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
8	7	sseq1i 3959	. 2 ⊢ ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷)
9		imassmpt.1	. . 3 ⊢ Ⅎ𝑥𝜑
10		eqid 2761	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
11		imassmpt.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉)
12	9, 10, 11	rnmptssbi 46241	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899   ↦ cmpt 5186  ran crn 5652   ↾ cres 5653   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6539  df-fn 6540  df-f 6541

This theorem is used by:  limsup10exlem  46751