Theorem rnmptssrn 45154

Description: Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
rnmptssrn.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
rnmptssrn.y	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐶 𝐵 = 𝐷)

Assertion

Ref	Expression
rnmptssrn	⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rnmptssrn

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = (𝑦 ∈ 𝐶 ↦ 𝐷)
2		rnmptssrn.y	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐶 𝐵 = 𝐷)
3		rnmptssrn.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
4	1, 2, 3	elrnmptd 5981	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
5	4	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
6		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	rnmptss 7150	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
8	5, 7	syl 17	1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3966   ↦ cmpt 5234  ran crn 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-fun 6571  df-fn 6572  df-f 6573

This theorem is referenced by:  sge0f1o  46366