Theorem rnmptssdf 45989

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptssdf

Step	Hyp	Ref	Expression
1		rnmptssdf.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rnmptssdf.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
3	1, 2	ralrimia 3264	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
4		rnmptssdf.2	. . 3 ⊢ Ⅎ𝑥𝐶
5		rnmptssdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	4, 5	rnmptssf 45982	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
7	3, 6	syl 18	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079   ⊆ wss 3905   ↦ cmpt 5192  ran crn 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  rnmptss2  45992  supminfrnmpt  46179  supminfxrrnmpt  46205