Theorem rnmptssbi 46035

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
rnmptssbi.1	⊢ Ⅎ𝑥𝜑
rnmptssbi.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rnmptssbi.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
rnmptssbi	⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

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

Proof of Theorem rnmptssbi

Step	Hyp	Ref	Expression
1		rnmptssbi.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		rnmptssbi.2	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfmpt1 5212	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	nfcxfr 2925	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5	4	nfrn 5944	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
6		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥𝐶
7	5, 6	nfss 3931	. . . 4 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐶
8	1, 7	nfan 1932	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ran 𝐹 ⊆ 𝐶)
9		simplr 781	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ 𝐶)
10		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
11		rnmptssbi.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
12	11	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
13	2, 10, 12	elrnmpt1d 5956	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
14	9, 13	sseldd 3939	. . 3 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
15	8, 14	ralrimia 3266	. 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
16	2	rnmptss 7122	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
17	16	adantl 487	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ran 𝐹 ⊆ 𝐶)
18	15, 17	impbida 813	1 ⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906   ↦ cmpt 5194  ran crn 5664

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  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 6542  df-fn 6543  df-f 6544

This theorem is used by:  imassmpt  46037