Theorem rnmptssd 45303

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 3231	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
4		rnmptssd.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	rnmptss 7056	. 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 1541  Ⅎwnf 1784   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897   ↦ cmpt 5170  ran crn 5615

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  infnsuprnmpt  45357  suprclrnmpt  45358  suprubrnmpt2  45359  suprubrnmpt  45360  fisupclrnmpt  45506  supxrleubrnmpt  45514  infxrlbrnmpt2  45518  supxrrernmpt  45529  suprleubrnmpt  45530  infrnmptle  45531  infxrunb3rnmpt  45536  supxrre3rnmpt  45537  supminfrnmpt  45553  infxrrnmptcl  45555  infxrgelbrnmpt  45562  infrpgernmpt  45573  supminfxrrnmpt  45579  liminfcl  45871  fourierdlem31  46246  fourierdlem53  46267  sge0xaddlem2  46542  sge0reuz  46555  sge0reuzb  46556  meadjiun  46574  hoidmvlelem2  46704  iunhoiioolem  46783  vonioolem1  46788  smflimsuplem4  46931