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Theorem ralssiun 34843
 Description: The index set of an indexed union is a subset of the union when each 𝐵 contains its index. (Contributed by ML, 16-Dec-2020.)
Assertion
Ref Expression
ralssiun (∀𝑥𝐴 𝑥𝐵𝐴 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ralssiun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfra1 3183 . 2 𝑥𝑥𝐴 𝑥𝐵
2 nfcv 2955 . 2 𝑥𝐴
3 nfiu1 4916 . 2 𝑥 𝑥𝐴 𝐵
4 simpr 488 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥𝐴)
5 rsp 3170 . . . . . . . . . . . . . 14 (∀𝑥𝐴 𝑥𝐵 → (𝑥𝐴𝑥𝐵))
65adantl 485 . . . . . . . . . . . . 13 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → (𝑥𝐴𝑥𝐵))
7 eleq1 2877 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
87imbi2d 344 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑦𝐵)))
98adantr 484 . . . . . . . . . . . . 13 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑦𝐵)))
106, 9mpbid 235 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → (𝑥𝐴𝑦𝐵))
1110imp 410 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
12 rspe 3263 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
134, 11, 12syl2anc 587 . . . . . . . . . 10 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → ∃𝑥𝐴 𝑦𝐵)
14 abid 2780 . . . . . . . . . 10 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ ∃𝑥𝐴 𝑦𝐵)
1513, 14sylibr 237 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
16 eleq1 2877 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}))
1716ad2antrr 725 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → (𝑥 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}))
1815, 17mpbird 260 . . . . . . . 8 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
19 df-iun 4884 . . . . . . . 8 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2018, 19eleqtrrdi 2901 . . . . . . 7 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥 𝑥𝐴 𝐵)
2120expl 461 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2221equcoms 2027 . . . . 5 (𝑦 = 𝑥 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2322vtocleg 3529 . . . 4 (𝑥𝐴 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2423anabsi7 670 . . 3 ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵)
2524ex 416 . 2 (∀𝑥𝐴 𝑥𝐵 → (𝑥𝐴𝑥 𝑥𝐴 𝐵))
261, 2, 3, 25ssrd 3920 1 (∀𝑥𝐴 𝑥𝐵𝐴 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∪ ciun 4882 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-iun 4884 This theorem is referenced by:  pibt2  34853
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