rnmptssrn - Mathbox for Glauco Siliprandi

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Theorem rnmptssrn 44549

Description: Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
rnmptssrn.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
rnmptssrn.y	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐶 𝐵 = 𝐷)

**Assertion**
Ref	Expression
rnmptssrn	⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))

Distinct variable groups: 𝑥,𝐴 𝑦,𝐵 𝑥,𝐶 𝑥,𝐷 𝜑,𝑥 𝑥,𝑦 Allowed substitution hints: 𝜑(𝑦) 𝐴(𝑦) 𝐵(𝑥) 𝐶(𝑦) 𝐷(𝑦) 𝑉(𝑥,𝑦)

**Proof of Theorem rnmptssrn**
Step	Hyp	Ref	Expression
1		eqid 2728	. . . 4 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = (𝑦 ∈ 𝐶 ↦ 𝐷)
2		rnmptssrn.y	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐶 𝐵 = 𝐷)
3		rnmptssrn.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
4	1, 2, 3	elrnmptd 5957	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
5	4	ralrimiva 3142	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
6		eqid 2728	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	rnmptss 7127	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ran (𝑦 ∈ 𝐶 ↦ 𝐷) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
8	5, 7	syl 17	1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 ⊆ wss 3945 ↦ cmpt 5225 ran crn 5673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-fun 6544 df-fn 6545 df-f 6546

This theorem is referenced by: sge0f1o 45764

	
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