Theorem rnmptssrn 46118

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 2760	. . . 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 5941	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
5	4	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
6		eqid 2760	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	rnmptss 7111	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
8	5, 7	syl 18	1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3898   ↦ cmpt 5185  ran crn 5648

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-fun 6529  df-fn 6530  df-f 6531

This theorem is used by:  sge0f1o  47314