Theorem iunxdif3 5095

Description: An indexed union where some terms are the empty set. See iunxdif2 5053. (Contributed by Thierry Arnoux, 4-May-2020.)

Hypothesis

Ref	Expression
iunxdif3.1	⊢ Ⅎ𝑥𝐸

Assertion

Ref	Expression
iunxdif3	⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐸(𝑥)

Proof of Theorem iunxdif3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 4238	. . . . . 6 ⊢ (𝐴 ∩ 𝐸) ⊆ 𝐸
2		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
3		iunxdif3.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐸
4	2, 3	nfin 4224	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐸)
5	4, 3	ssrexf 4050	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐸) ⊆ 𝐸 → (∃𝑥 ∈ (𝐴 ∩ 𝐸)𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐸 𝑦 ∈ 𝐵))
6		eliun 4995	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐸)𝑦 ∈ 𝐵)
7		eliun 4995	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐸 𝐵 ↔ ∃𝑥 ∈ 𝐸 𝑦 ∈ 𝐵)
8	5, 6, 7	3imtr4g 296	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐸) ⊆ 𝐸 → (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐸 𝐵))
9	8	ssrdv 3989	. . . . . 6 ⊢ ((𝐴 ∩ 𝐸) ⊆ 𝐸 → ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ⊆ ∪ 𝑥 ∈ 𝐸 𝐵)
10	1, 9	ax-mp 5	. . . . 5 ⊢ ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ⊆ ∪ 𝑥 ∈ 𝐸 𝐵
11		iuneq2 5011	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐸 𝐵 = ∪ 𝑥 ∈ 𝐸 ∅)
12		iun0 5062	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐸 ∅ = ∅
13	11, 12	eqtrdi 2793	. . . . 5 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐸 𝐵 = ∅)
14	10, 13	sseqtrid 4026	. . . 4 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ⊆ ∅)
15		ss0 4402	. . . 4 ⊢ (∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ⊆ ∅ → ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 = ∅)
16	14, 15	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 = ∅)
17	16	uneq1d 4167	. 2 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → (∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = (∅ ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵))
18		iunxun 5094	. . . 4 ⊢ ∪ 𝑥 ∈ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))𝐵 = (∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵)
19		inundif 4479	. . . . 5 ⊢ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴
20	19	nfth 1801	. . . . . 6 ⊢ Ⅎ𝑥((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴
21	2, 3	nfdif 4129	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐸)
22	4, 21	nfun 4170	. . . . . 6 ⊢ Ⅎ𝑥((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))
23		id 22	. . . . . 6 ⊢ (((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴 → ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴)
24		eqidd 2738	. . . . . 6 ⊢ (((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴 → 𝐵 = 𝐵)
25	20, 22, 2, 23, 24	iuneq12df 5018	. . . . 5 ⊢ (((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴 → ∪ 𝑥 ∈ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
26	19, 25	ax-mp 5	. . . 4 ⊢ ∪ 𝑥 ∈ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵
27	18, 26	eqtr3i 2767	. . 3 ⊢ (∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵
28	27	a1i 11	. 2 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → (∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
29		uncom 4158	. . . 4 ⊢ (∅ ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = (∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 ∪ ∅)
30		un0 4394	. . . 4 ⊢ (∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 ∪ ∅) = ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵
31	29, 30	eqtri 2765	. . 3 ⊢ (∅ ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵
32	31	a1i 11	. 2 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → (∅ ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵)
33	17, 28, 32	3eqtr3rd 2786	1 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ∪ ciun 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-iun 4993

This theorem is referenced by:  aciunf1  32673  suppovss  32690  ovnsubadd2lem  46660