Theorem rnmptssrn 45493

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 5913	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
5	4	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
6		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	rnmptss 7070	. 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 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061   ⊆ wss 3902   ↦ cmpt 5180  ran crn 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6495  df-fn 6496  df-f 6497

This theorem is referenced by:  sge0f1o  46693