Theorem rnmptssbi 46545

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 5210	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	nfcxfr 2924	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5	4	nfrn 5946	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
6		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝐶
7	5, 6	nfss 3906	. . . 4 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐶
8	1, 7	nfan 1934	. . 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 5958	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
14	9, 13	sseldd 3914	. . 3 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
15	8, 14	ralrimia 3265	. 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
16	2	rnmptss 7155	. . 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 2147  ∀wral 3080   ⊆ wss 3900   ↦ cmpt 5192  ran crn 5659

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-f 6570

This theorem is used by:  imassmpt  46547