Theorem rnmptssbi 41554

Description: The range of an operation 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 5164	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	nfcxfr 2975	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5	4	nfrn 5824	. . . . 5 ⊢ Ⅎ𝑥ran 𝐹
6		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥𝐶
7	5, 6	nfss 3960	. . . 4 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐶
8	1, 7	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ran 𝐹 ⊆ 𝐶)
9		simplr 767	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ 𝐶)
10		simpr 487	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
11		rnmptssbi.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
12	11	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
13	2, 10, 12	elrnmpt1d 41520	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹)
14	9, 13	sseldd 3968	. . 3 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
15	8, 14	ralrimia 41418	. 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
16	2	rnmptss 6886	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
17	16	adantl 484	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ran 𝐹 ⊆ 𝐶)
18	15, 17	impbida 799	1 ⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936   ↦ cmpt 5146  ran crn 5556

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363

This theorem is referenced by:  imassmpt  41557