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Theorem iunxdif3 4758
 Description: An indexed union where some terms are the empty set. See iunxdif2 4720. (Contributed by Thierry Arnoux, 4-May-2020.)
Hypothesis
Ref Expression
iunxdif3.1 𝑥𝐸
Assertion
Ref Expression
iunxdif3 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 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.
StepHypRef Expression
1 inss2 3977 . . . . . 6 (𝐴𝐸) ⊆ 𝐸
2 nfcv 2902 . . . . . . . . . 10 𝑥𝐴
3 iunxdif3.1 . . . . . . . . . 10 𝑥𝐸
42, 3nfin 3963 . . . . . . . . 9 𝑥(𝐴𝐸)
54, 3ssrexf 3806 . . . . . . . 8 ((𝐴𝐸) ⊆ 𝐸 → (∃𝑥 ∈ (𝐴𝐸)𝑦𝐵 → ∃𝑥𝐸 𝑦𝐵))
6 eliun 4676 . . . . . . . 8 (𝑦 𝑥 ∈ (𝐴𝐸)𝐵 ↔ ∃𝑥 ∈ (𝐴𝐸)𝑦𝐵)
7 eliun 4676 . . . . . . . 8 (𝑦 𝑥𝐸 𝐵 ↔ ∃𝑥𝐸 𝑦𝐵)
85, 6, 73imtr4g 285 . . . . . . 7 ((𝐴𝐸) ⊆ 𝐸 → (𝑦 𝑥 ∈ (𝐴𝐸)𝐵𝑦 𝑥𝐸 𝐵))
98ssrdv 3750 . . . . . 6 ((𝐴𝐸) ⊆ 𝐸 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵)
101, 9ax-mp 5 . . . . 5 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵
11 iuneq2 4689 . . . . . 6 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = 𝑥𝐸 ∅)
12 iun0 4728 . . . . . 6 𝑥𝐸 ∅ = ∅
1311, 12syl6eq 2810 . . . . 5 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = ∅)
1410, 13syl5sseq 3794 . . . 4 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅)
15 ss0 4117 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1614, 15syl 17 . . 3 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1716uneq1d 3909 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵))
18 iunxun 4757 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵)
19 inundif 4190 . . . . 5 ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
2019nfth 1876 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
212, 3nfdif 3874 . . . . . . 7 𝑥(𝐴𝐸)
224, 21nfun 3912 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸))
23 id 22 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 → ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴)
24 eqidd 2761 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴𝐵 = 𝐵)
2520, 22, 2, 23, 24iuneq12df 4696 . . . . 5 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵)
2619, 25ax-mp 5 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵
2718, 26eqtr3i 2784 . . 3 ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵
2827a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵)
29 uncom 3900 . . . 4 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅)
30 un0 4110 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅) = 𝑥 ∈ (𝐴𝐸)𝐵
3129, 30eqtri 2782 . . 3 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵
3231a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵)
3317, 28, 323eqtr3rd 2803 1 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Ⅎwnfc 2889  ∀wral 3050  ∃wrex 3051   ∖ cdif 3712   ∪ cun 3713   ∩ cin 3714   ⊆ wss 3715  ∅c0 4058  ∪ ciun 4672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-iun 4674 This theorem is referenced by:  aciunf1  29772  ovnsubadd2lem  41365
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