Theorem rnmptssd 7076

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 3236	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
4		rnmptssd.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 7075	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
6	3, 5	syl 17	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889   ↦ cmpt 5166  ran crn 5632

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500  df-fn 6501  df-f 6502

This theorem is referenced by:  esplyfval1  33717  esplyfvaln  33718  infnsuprnmpt  45679  suprclrnmpt  45680  suprubrnmpt2  45681  suprubrnmpt  45682  fisupclrnmpt  45827  supxrleubrnmpt  45834  infxrlbrnmpt2  45838  supxrrernmpt  45849  suprleubrnmpt  45850  infrnmptle  45851  infxrunb3rnmpt  45856  supxrre3rnmpt  45857  supminfrnmpt  45873  infxrrnmptcl  45875  infxrgelbrnmpt  45882  infrpgernmpt  45893  supminfxrrnmpt  45899  liminfcl  46191  fourierdlem31  46566  fourierdlem53  46587  sge0xaddlem2  46862  sge0reuz  46875  sge0reuzb  46876  meadjiun  46894  hoidmvlelem2  47024  iunhoiioolem  47103  vonioolem1  47108  smflimsuplem4  47251