Theorem rnmptssrn 41808

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 2798	. . . 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 5797	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
5	4	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
6		eqid 2798	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	rnmptss 6863	. 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   ↦ cmpt 5110  ran crn 5520

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332

This theorem is referenced by:  sge0f1o  43021