Theorem iunrnmptss 30100

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 3088	. . . 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 3126	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
4		eqid 2772	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	elrnmptg 5671	. . . . . . . 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 620	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶)))
8	7	exbidv 1880	. . . . 5 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶)))
9		r19.41v 3282	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶))
10		iunrnmptss.1	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
11	10	eleq2d 2845	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
12	11	biimpa 469	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐷)
13	12	reximi 3184	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
14	9, 13	sylbir 227	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
15	14	exlimiv 1889	. . . . 5 ⊢ (∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷)
16	8, 15	syl6bi 245	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑧 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷))
17	1, 16	syl5bi 234	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷))
18	17	ss2abdv 3928	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶} ⊆ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷})
19		df-iun 4790	. 2 ⊢ ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 = {𝑧 ∣ ∃𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ∈ 𝐶}
20		df-iun 4790	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐷 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐷}
21	18, 19, 20	3sstr4g 3896	1 ⊢ (𝜑 → ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐷)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  ∃wex 1742   ∈ wcel 2050  {cab 2752  ∀wral 3082  ∃wrex 3083   ⊆ wss 3823  ∪ ciun 4788   ↦ cmpt 5004  ran crn 5404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-cnv 5411  df-dm 5413  df-rn 5414

This theorem is referenced by:  fnpreimac  30192