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Theorem eliuniin2 39818
Description: Indexed union of indexed intersections. See eliincex 39808 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
eliuniin2.1 𝑥𝐶
eliuniin2.2 𝐴 = 𝑥𝐵 𝑦𝐶 𝐷
Assertion
Ref Expression
eliuniin2 (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐶   𝑥,𝑍   𝑦,𝑍
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥)   𝐷(𝑥,𝑦)

Proof of Theorem eliuniin2
StepHypRef Expression
1 eliuniin2.2 . . . . 5 𝐴 = 𝑥𝐵 𝑦𝐶 𝐷
21eleq2i 2841 . . . 4 (𝑍𝐴𝑍 𝑥𝐵 𝑦𝐶 𝐷)
3 eliun 4656 . . . 4 (𝑍 𝑥𝐵 𝑦𝐶 𝐷 ↔ ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
42, 3sylbb 209 . . 3 (𝑍𝐴 → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
5 eliin 4657 . . . . . 6 (𝑍 𝑦𝐶 𝐷 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
65ibi 256 . . . . 5 (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷)
76a1i 11 . . . 4 (𝑍𝐴 → (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷))
87reximdv 3163 . . 3 (𝑍𝐴 → (∃𝑥𝐵 𝑍 𝑦𝐶 𝐷 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
94, 8mpd 15 . 2 (𝑍𝐴 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
10 eliuniin2.1 . . . 4 𝑥𝐶
11 nfcv 2912 . . . 4 𝑥
1210, 11nfne 3042 . . 3 𝑥 𝐶 ≠ ∅
13 nfv 1994 . . 3 𝑥 𝑍𝐴
14 simp2 1130 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑥𝐵)
15 eliin2 39814 . . . . . . . 8 (𝐶 ≠ ∅ → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
1615biimpar 463 . . . . . . 7 ((𝐶 ≠ ∅ ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑦𝐶 𝐷)
17 rspe 3150 . . . . . . 7 ((𝑥𝐵𝑍 𝑦𝐶 𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1814, 16, 173imp3i2an 1435 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1918, 3sylibr 224 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑥𝐵 𝑦𝐶 𝐷)
2019, 2sylibr 224 . . . 4 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍𝐴)
21203exp 1111 . . 3 (𝐶 ≠ ∅ → (𝑥𝐵 → (∀𝑦𝐶 𝑍𝐷𝑍𝐴)))
2212, 13, 21rexlimd 3173 . 2 (𝐶 ≠ ∅ → (∃𝑥𝐵𝑦𝐶 𝑍𝐷𝑍𝐴))
239, 22impbid2 216 1 (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1070   = wceq 1630  wcel 2144  wnfc 2899  wne 2942  wral 3060  wrex 3061  c0 4061   ciun 4652   ciin 4653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-nul 4062  df-iun 4654  df-iin 4655
This theorem is referenced by: (None)
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