Theorem rnmptssd 7145

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 3265	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
4		rnmptssd.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 7144	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
6	3, 5	syl 18	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ↦ cmpt 5189  ran crn 5656

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6557  df-fn 6558  df-f 6559

This theorem is used by:  esplyfval1  34272  esplyfvaln  34273  infnsuprnmpt  46465  suprclrnmpt  46466  suprubrnmpt2  46467  suprubrnmpt  46468  fisupclrnmpt  46612  supxrleubrnmpt  46619  infxrlbrnmpt2  46623  supxrrernmpt  46634  suprleubrnmpt  46635  infrnmptle  46636  infxrunb3rnmpt  46641  supxrre3rnmpt  46642  supminfrnmpt  46658  infxrrnmptcl  46660  infxrgelbrnmpt  46667  infrpgernmpt  46678  supminfxrrnmpt  46684  liminfcl  46976  fourierdlem31  47351  fourierdlem53  47372  sge0xaddlem2  47647  sge0reuz  47660  sge0reuzb  47661  meadjiun  47679  hoidmvlelem2  47809  iunhoiioolem  47888  vonioolem1  47893  smflimsuplem4  48036