Theorem ss2iundf 44103

Description: Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)

Hypotheses

Ref	Expression
ss2iundf.xph	⊢ Ⅎ𝑥𝜑
ss2iundf.yph	⊢ Ⅎ𝑦𝜑
ss2iundf.y	⊢ Ⅎ𝑦𝑌
ss2iundf.a	⊢ Ⅎ𝑦𝐴
ss2iundf.b	⊢ Ⅎ𝑦𝐵
ss2iundf.xc	⊢ Ⅎ𝑥𝐶
ss2iundf.yc	⊢ Ⅎ𝑦𝐶
ss2iundf.d	⊢ Ⅎ𝑥𝐷
ss2iundf.g	⊢ Ⅎ𝑦𝐺
ss2iundf.el	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
ss2iundf.sub	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
ss2iundf.ss	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)

Assertion

Ref	Expression
ss2iundf	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem ss2iundf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ss2iundf.xph	. . 3 ⊢ Ⅎ𝑥𝜑
2		ss2iundf.ss	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)
3		df-ral 3054	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐶 ¬ 𝐵 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷))
4		ss2iundf.el	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
5		ss2iundf.yph	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜑
6		ss2iundf.a	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝐴
7	6	nfcri 2893	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
8	5, 7	nfan 1906	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴)
9		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
10	9	eleq1d 2824	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = 𝑌) → (𝑦 ∈ 𝐶 ↔ 𝑌 ∈ 𝐶))
11	10	biimprd 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = 𝑌) → (𝑌 ∈ 𝐶 → 𝑦 ∈ 𝐶))
12		ss2iundf.sub	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
13	12	sseq2d 3947	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → (𝐵 ⊆ 𝐷 ↔ 𝐵 ⊆ 𝐺))
14	13	3expa 1124	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = 𝑌) → (𝐵 ⊆ 𝐷 ↔ 𝐵 ⊆ 𝐺))
15	14	notbid 319	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = 𝑌) → (¬ 𝐵 ⊆ 𝐷 ↔ ¬ 𝐵 ⊆ 𝐺))
16	15	biimpd 230	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = 𝑌) → (¬ 𝐵 ⊆ 𝐷 → ¬ 𝐵 ⊆ 𝐺))
17	11, 16	imim12d 81	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = 𝑌) → ((𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷) → (𝑌 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐺)))
18	17	ex 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝑌 → ((𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷) → (𝑌 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐺))))
19	8, 18	alrimi 2225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 = 𝑌 → ((𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷) → (𝑌 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐺))))
20		ss2iundf.y	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝑌
21		ss2iundf.yc	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝐶
22	20, 21	nfel 2915	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑌 ∈ 𝐶
23		ss2iundf.b	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝐵
24		ss2iundf.g	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝐺
25	23, 24	nfss 3908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝐵 ⊆ 𝐺
26	25	nfn 1864	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 ¬ 𝐵 ⊆ 𝐺
27	22, 26	nfim 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑌 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐺)
28	27, 20	spcimgfi1 3493	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑌 → ((𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷) → (𝑌 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐺))) → (𝑌 ∈ 𝐶 → (∀𝑦(𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷) → (𝑌 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐺))))
29	19, 4, 28	sylc 65	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦(𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷) → (𝑌 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐺)))
30	4, 29	mpid 44	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦(𝑦 ∈ 𝐶 → ¬ 𝐵 ⊆ 𝐷) → ¬ 𝐵 ⊆ 𝐺))
31	3, 30	biimtrid 243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐶 ¬ 𝐵 ⊆ 𝐷 → ¬ 𝐵 ⊆ 𝐺))
32	31	con2d 134	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ 𝐺 → ¬ ∀𝑦 ∈ 𝐶 ¬ 𝐵 ⊆ 𝐷))
33		dfrex2 3066	. . . . 5 ⊢ (∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷 ↔ ¬ ∀𝑦 ∈ 𝐶 ¬ 𝐵 ⊆ 𝐷)
34	32, 33	imbitrrdi 253	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ 𝐺 → ∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷))
35	2, 34	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷)
36	1, 35	ralrimia 3238	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷)
37		ssel 3909	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐷 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐷))
38	37	reximi 3077	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷 → ∃𝑦 ∈ 𝐶 (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐷))
39	23	nfcri 2893	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
40	39	r19.37 3242	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐷) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
41	38, 40	syl 17	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷 → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
42		eliun 4925	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐶 𝐷 ↔ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)
43	41, 42	imbitrrdi 253	. . . . 5 ⊢ (∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷 → (𝑧 ∈ 𝐵 → 𝑧 ∈ ∪ 𝑦 ∈ 𝐶 𝐷))
44	43	ssrdv 3921	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷 → 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
45	44	ralimi 3076	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
46		ss2iundf.xc	. . . . 5 ⊢ Ⅎ𝑥𝐶
47		ss2iundf.d	. . . . 5 ⊢ Ⅎ𝑥𝐷
48	46, 47	nfiun 4953	. . . 4 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐶 𝐷
49	48	iunssf 4972	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
50	45, 49	sylibr 235	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝐵 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
51	36, 50	syl 17	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092  ∀wal 1545   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  Ⅎwnfc 2886  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-v 3433  df-ss 3900  df-iun 4923

This theorem is referenced by:  ss2iundv  44104  cbviuneq12df  44105