Theorem rnmptssd 7123

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
rnmptssd.1	⊢ Ⅎ𝑥𝜑
rnmptssd.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rnmptssd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
rnmptssd	⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptssd

Step	Hyp	Ref	Expression
1		rnmptssd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rnmptssd.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
3	1, 2	ralrimia 3271	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
4		rnmptssd.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 7122	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
6	3, 5	syl 18	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568  Ⅎwnf 1811   ∈ wcel 2150  ∀wral 3086   ⊆ wss 3913   ↦ cmpt 5197  ran crn 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-fun 6542  df-fn 6543  df-f 6544

This theorem is referenced by:  esplyfval1  33933  esplyfvaln  33934  infnsuprnmpt  45917  suprclrnmpt  45918  suprubrnmpt2  45919  suprubrnmpt  45920  fisupclrnmpt  46065  supxrleubrnmpt  46072  infxrlbrnmpt2  46076  supxrrernmpt  46087  suprleubrnmpt  46088  infrnmptle  46089  infxrunb3rnmpt  46094  supxrre3rnmpt  46095  supminfrnmpt  46111  infxrrnmptcl  46113  infxrgelbrnmpt  46120  infrpgernmpt  46131  supminfxrrnmpt  46137  liminfcl  46429  fourierdlem31  46804  fourierdlem53  46825  sge0xaddlem2  47100  sge0reuz  47113  sge0reuzb  47114  meadjiun  47132  hoidmvlelem2  47262  iunhoiioolem  47341  vonioolem1  47346  smflimsuplem4  47489