Theorem rnmptssdff 45254

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

Hypotheses

Ref	Expression
rnmptssdff.1	⊢ Ⅎ𝑥𝜑
rnmptssdff.2	⊢ Ⅎ𝑥𝐴
rnmptssdff.3	⊢ Ⅎ𝑥𝐶
rnmptssdff.4	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rnmptssdff.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Assertion

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

Proof of Theorem rnmptssdff

Step	Hyp	Ref	Expression
1		rnmptssdff.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rnmptssdff.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
3	1, 2	ralrimia 3244	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
4		rnmptssdff.2	. . 3 ⊢ Ⅎ𝑥𝐴
5		rnmptssdff.3	. . 3 ⊢ Ⅎ𝑥𝐶
6		rnmptssdff.4	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	4, 5, 6	rnmptssff 45253	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
8	3, 7	syl 17	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  Ⅎwnf 1782   ∈ wcel 2107  Ⅎwnfc 2882  ∀wral 3050   ⊆ wss 3931   ↦ cmpt 5205  ran crn 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  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 6543  df-fn 6544  df-f 6545

This theorem is referenced by:  saliunclf  46309