Theorem rnmptss2 46000

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
rnmptss2.1	⊢ Ⅎ𝑥𝜑
rnmptss2.3	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
rnmptss2.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
rnmptss2	⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem rnmptss2

Step	Hyp	Ref	Expression
1		rnmptss2.1	. 2 ⊢ Ⅎ𝑥𝜑
2		nfmpt1 5209	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ 𝐶)
3	2	nfrn 5941	. 2 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐵 ↦ 𝐶)
4		eqid 2762	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
5		eqid 2762	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
6		rnmptss2.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	6	sselda 3936	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
8		rnmptss2.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
9	5, 7, 8	elrnmpt1d 5953	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶))
10	1, 3, 4, 9	rnmptssdf 45997	1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  Ⅎwnf 1812   ∈ wcel 2142   ⊆ wss 3904   ↦ cmpt 5191  ran crn 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-fun 6538  df-fn 6539  df-f 6540

This theorem is used by:  smflimsuplem4  47565