Theorem rnmptss2 44260

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 5256	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ 𝐶)
3	2	nfrn 5951	. 2 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐵 ↦ 𝐶)
4		eqid 2731	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
5		eqid 2731	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
6		rnmptss2.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	6	sselda 3982	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
8		rnmptss2.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
9	5, 7, 8	elrnmpt1d 44231	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶))
10	1, 3, 4, 9	rnmptssdf 44257	1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2105   ⊆ wss 3948   ↦ cmpt 5231  ran crn 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547

This theorem is referenced by:  smflimsuplem4  45838