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Theorem ralssiun 36592
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 3280 . 2 𝑥𝑥𝐴 𝑥𝐵
2 nfcv 2902 . 2 𝑥𝐴
3 nfiu1 5031 . 2 𝑥 𝑥𝐴 𝐵
4 simpr 484 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥𝐴)
5 rsp 3243 . . . . . . . . . . . . . 14 (∀𝑥𝐴 𝑥𝐵 → (𝑥𝐴𝑥𝐵))
65adantl 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → (𝑥𝐴𝑥𝐵))
7 eleq1 2820 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
87imbi2d 340 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑦𝐵)))
98adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑦𝐵)))
106, 9mpbid 231 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → (𝑥𝐴𝑦𝐵))
1110imp 406 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
12 rspe 3245 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
134, 11, 12syl2anc 583 . . . . . . . . . 10 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → ∃𝑥𝐴 𝑦𝐵)
14 abid 2712 . . . . . . . . . 10 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ ∃𝑥𝐴 𝑦𝐵)
1513, 14sylibr 233 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
16 eleq1 2820 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}))
1716ad2antrr 723 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → (𝑥 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}))
1815, 17mpbird 257 . . . . . . . 8 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
19 df-iun 4999 . . . . . . . 8 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2018, 19eleqtrrdi 2843 . . . . . . 7 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥 𝑥𝐴 𝐵)
2120expl 457 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2221equcoms 2022 . . . . 5 (𝑦 = 𝑥 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2322vtocleg 3541 . . . 4 (𝑥𝐴 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2423anabsi7 668 . . 3 ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵)
2524ex 412 . 2 (∀𝑥𝐴 𝑥𝐵 → (𝑥𝐴𝑥 𝑥𝐴 𝐵))
261, 2, 3, 25ssrd 3987 1 (∀𝑥𝐴 𝑥𝐵𝐴 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wral 3060  wrex 3069  wss 3948   ciun 4997
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 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965  df-iun 4999
This theorem is referenced by:  pibt2  36602
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