Theorem rnmptssff 46017

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
rnmptssff.1	⊢ Ⅎ𝑥𝐴
rnmptssff.2	⊢ Ⅎ𝑥𝐶
rnmptssff.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
rnmptssff	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Proof of Theorem rnmptssff

Step	Hyp	Ref	Expression
1		rnmptssff.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		rnmptssff.2	. . 3 ⊢ Ⅎ𝑥𝐶
3		rnmptssff.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	1, 2, 3	fmptff 46012	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
5		frn 6713	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
6	4, 5	sylbi 220	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079   ⊆ wss 3905   ↦ cmpt 5192  ran crn 5662  ⟶wf 6532

This proof depends on 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 proof 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 used by:  rnmptssdff  46018