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Theorem iunxdif3 5039
Description: An indexed union where some terms are the empty set. See iunxdif2 4997. (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 4175 . . . . . 6 (𝐴𝐸) ⊆ 𝐸
2 nfcv 2904 . . . . . . . . . 10 𝑥𝐴
3 iunxdif3.1 . . . . . . . . . 10 𝑥𝐸
42, 3nfin 4162 . . . . . . . . 9 𝑥(𝐴𝐸)
54, 3ssrexf 3995 . . . . . . . 8 ((𝐴𝐸) ⊆ 𝐸 → (∃𝑥 ∈ (𝐴𝐸)𝑦𝐵 → ∃𝑥𝐸 𝑦𝐵))
6 eliun 4942 . . . . . . . 8 (𝑦 𝑥 ∈ (𝐴𝐸)𝐵 ↔ ∃𝑥 ∈ (𝐴𝐸)𝑦𝐵)
7 eliun 4942 . . . . . . . 8 (𝑦 𝑥𝐸 𝐵 ↔ ∃𝑥𝐸 𝑦𝐵)
85, 6, 73imtr4g 295 . . . . . . 7 ((𝐴𝐸) ⊆ 𝐸 → (𝑦 𝑥 ∈ (𝐴𝐸)𝐵𝑦 𝑥𝐸 𝐵))
98ssrdv 3937 . . . . . 6 ((𝐴𝐸) ⊆ 𝐸 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵)
101, 9ax-mp 5 . . . . 5 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵
11 iuneq2 4957 . . . . . 6 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = 𝑥𝐸 ∅)
12 iun0 5006 . . . . . 6 𝑥𝐸 ∅ = ∅
1311, 12eqtrdi 2792 . . . . 5 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = ∅)
1410, 13sseqtrid 3983 . . . 4 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅)
15 ss0 4344 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1614, 15syl 17 . . 3 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1716uneq1d 4108 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵))
18 iunxun 5038 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵)
19 inundif 4424 . . . . 5 ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
2019nfth 1802 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
212, 3nfdif 4071 . . . . . . 7 𝑥(𝐴𝐸)
224, 21nfun 4111 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸))
23 id 22 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 → ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴)
24 eqidd 2737 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴𝐵 = 𝐵)
2520, 22, 2, 23, 24iuneq12df 4964 . . . . 5 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵)
2619, 25ax-mp 5 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵
2718, 26eqtr3i 2766 . . 3 ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵
2827a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵)
29 uncom 4099 . . . 4 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅)
30 un0 4336 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅) = 𝑥 ∈ (𝐴𝐸)𝐵
3129, 30eqtri 2764 . . 3 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵
3231a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵)
3317, 28, 323eqtr3rd 2785 1 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wnfc 2884  wral 3061  wrex 3070  cdif 3894  cun 3895  cin 3896  wss 3897  c0 4268   ciun 4938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-iun 4940
This theorem is referenced by:  aciunf1  31200  suppovss  31217  ovnsubadd2lem  44509
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