MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunxdif3 Structured version   Visualization version   GIF version

Theorem iunxdif3 4536
Description: An indexed union where some terms are the empty set. See iunxdif2 4498. (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 3795 . . . . . 6 (𝐴𝐸) ⊆ 𝐸
2 nfcv 2750 . . . . . . . . . 10 𝑥𝐴
3 iunxdif3.1 . . . . . . . . . 10 𝑥𝐸
42, 3nfin 3781 . . . . . . . . 9 𝑥(𝐴𝐸)
54, 3ssrexf 3627 . . . . . . . 8 ((𝐴𝐸) ⊆ 𝐸 → (∃𝑥 ∈ (𝐴𝐸)𝑦𝐵 → ∃𝑥𝐸 𝑦𝐵))
6 eliun 4454 . . . . . . . 8 (𝑦 𝑥 ∈ (𝐴𝐸)𝐵 ↔ ∃𝑥 ∈ (𝐴𝐸)𝑦𝐵)
7 eliun 4454 . . . . . . . 8 (𝑦 𝑥𝐸 𝐵 ↔ ∃𝑥𝐸 𝑦𝐵)
85, 6, 73imtr4g 283 . . . . . . 7 ((𝐴𝐸) ⊆ 𝐸 → (𝑦 𝑥 ∈ (𝐴𝐸)𝐵𝑦 𝑥𝐸 𝐵))
98ssrdv 3573 . . . . . 6 ((𝐴𝐸) ⊆ 𝐸 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵)
101, 9ax-mp 5 . . . . 5 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵
11 iuneq2 4467 . . . . . 6 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = 𝑥𝐸 ∅)
12 iun0 4506 . . . . . 6 𝑥𝐸 ∅ = ∅
1311, 12syl6eq 2659 . . . . 5 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = ∅)
1410, 13syl5sseq 3615 . . . 4 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅)
15 ss0 3925 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1614, 15syl 17 . . 3 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1716uneq1d 3727 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵))
18 iunxun 4535 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵)
19 inundif 3997 . . . . 5 ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
2019nfth 1717 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
212, 3nfdif 3692 . . . . . . 7 𝑥(𝐴𝐸)
224, 21nfun 3730 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸))
23 id 22 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 → ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴)
24 eqidd 2610 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴𝐵 = 𝐵)
2520, 22, 2, 23, 24iuneq12df 4474 . . . . 5 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵)
2619, 25ax-mp 5 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵
2718, 26eqtr3i 2633 . . 3 ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵
2827a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵)
29 uncom 3718 . . . 4 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅)
30 un0 3918 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅) = 𝑥 ∈ (𝐴𝐸)𝐵
3129, 30eqtri 2631 . . 3 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵
3231a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵)
3317, 28, 323eqtr3rd 2652 1 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wnfc 2737  wral 2895  wrex 2896  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873   ciun 4449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-iun 4451
This theorem is referenced by:  aciunf1  28651  ovnsubadd2lem  39339
  Copyright terms: Public domain W3C validator