Theorem rnmptssdff 45174

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 3264	. 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 45173	. 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 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893  ∀wral 3067   ⊆ wss 3976   ↦ cmpt 5249  ran crn 5696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-fun 6570  df-fn 6571  df-f 6572

This theorem is referenced by:  saliunclf  46232