Theorem rnmptssdff 45733

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 3240	. 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 45732	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
8	3, 7	syl 17	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  Ⅎwnfc 2888  ∀wral 3055   ⊆ wss 3885   ↦ cmpt 5156  ran crn 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6491  df-fn 6492  df-f 6493

This theorem is referenced by:  saliunclf  46779