Theorem iunxdif3 4542

Description: An indexed union where some terms are the empty set. See iunxdif2 4504. (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 3796	. . . . . 6 ⊢ (𝐴 ∩ 𝐸) ⊆ 𝐸
2		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
3		iunxdif3.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐸
4	2, 3	nfin 3782	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐸)
5	4, 3	ssrexf 3628	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐸) ⊆ 𝐸 → (∃𝑥 ∈ (𝐴 ∩ 𝐸)𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐸 𝑦 ∈ 𝐵))
6		eliun 4460	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐸)𝑦 ∈ 𝐵)
7		eliun 4460	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐸 𝐵 ↔ ∃𝑥 ∈ 𝐸 𝑦 ∈ 𝐵)
8	5, 6, 7	3imtr4g 284	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐸) ⊆ 𝐸 → (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐸 𝐵))
9	8	ssrdv 3574	. . . . . 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 4473	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐸 𝐵 = ∪ 𝑥 ∈ 𝐸 ∅)
12		iun0 4512	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐸 ∅ = ∅
13	11, 12	syl6eq 2660	. . . . 5 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐸 𝐵 = ∅)
14	10, 13	syl5sseq 3616	. . . 4 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ⊆ ∅)
15		ss0 3926	. . . 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 3728	. 2 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → (∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = (∅ ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵))
18		iunxun 4541	. . . 4 ⊢ ∪ 𝑥 ∈ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))𝐵 = (∪ 𝑥 ∈ (𝐴 ∩ 𝐸)𝐵 ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵)
19		inundif 3998	. . . . 5 ⊢ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴
20	19	nfth 1718	. . . . . 6 ⊢ Ⅎ𝑥((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴
21	2, 3	nfdif 3693	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐸)
22	4, 21	nfun 3731	. . . . . 6 ⊢ Ⅎ𝑥((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))
23		id 22	. . . . . 6 ⊢ (((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴 → ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴)
24		eqidd 2611	. . . . . 6 ⊢ (((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴 → 𝐵 = 𝐵)
25	20, 22, 2, 23, 24	iuneq12df 4480	. . . . 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 2634	. . 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 3719	. . . 4 ⊢ (∅ ∪ ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵) = (∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 ∪ ∅)
30		un0 3919	. . . 4 ⊢ (∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 ∪ ∅) = ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵
31	29, 30	eqtri 2632	. . 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 2653	1 ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-iun 4457

This theorem is referenced by:  aciunf1  28845  ovnsubadd2lem  39535