Theorem iunrnmptss 30806

Description: A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.)

Hypotheses

Ref	Expression
iunrnmptss.1	⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
iunrnmptss.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
iunrnmptss	⊢ (𝜑 → ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐷)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐶   𝑦,𝐷   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem iunrnmptss

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3069	. . . 4 ⊢ (∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶 ↔ ∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶))
2		iunrnmptss.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
3	2	ralrimiva 3107	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
4		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	elrnmptg 5857	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
7	6	anbi1d 629	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶)))
8	7	exbidv 1925	. . . . 5 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶)))
9		r19.41v 3273	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶))
10		iunrnmptss.1	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
11	10	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
12	11	biimpa 476	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐷)
13	12	reximi 3174	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
14	9, 13	sylbir 234	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
15	14	exlimiv 1934	. . . . 5 ⊢ (∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
16	8, 15	syl6bi 252	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷))
17	1, 16	syl5bi 241	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷))
18	17	ss2abdv 3993	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶} ⊆ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷})
19		df-iun 4923	. 2 ⊢ ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 = {𝑧 ∣ ∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶}
20		df-iun 4923	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐷 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷}
21	18, 19, 20	3sstr4g 3962	1 ⊢ (𝜑 → ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐷)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∪ ciun 4921   ↦ cmpt 5153  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  fnpreimac  30910