Theorem rnmptssbi 45367

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 5188	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	nfcxfr 2892	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5	4	nfrn 5891	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
6		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥𝐶
7	5, 6	nfss 3922	. . . 4 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐶
8	1, 7	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ran 𝐹 ⊆ 𝐶)
9		simplr 768	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ 𝐶)
10		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
11		rnmptssbi.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
12	11	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
13	2, 10, 12	elrnmpt1d 5903	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
14	9, 13	sseldd 3930	. . 3 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
15	8, 14	ralrimia 3231	. 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
16	2	rnmptss 7056	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
17	16	adantl 481	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ran 𝐹 ⊆ 𝐶)
18	15, 17	impbida 800	1 ⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897   ↦ cmpt 5170  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  imassmpt  45369