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Theorem ralssiun 37390
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 3282 . 2 𝑥𝑥𝐴 𝑥𝐵
2 nfcv 2903 . 2 𝑥𝐴
3 nfiu1 5032 . 2 𝑥 𝑥𝐴 𝐵
4 simpr 484 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥𝐴)
5 rsp 3245 . . . . . . . . . . . . . 14 (∀𝑥𝐴 𝑥𝐵 → (𝑥𝐴𝑥𝐵))
65adantl 481 . . . . . . . . . . . . 13 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → (𝑥𝐴𝑥𝐵))
7 eleq1 2827 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
87imbi2d 340 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑦𝐵)))
98adantr 480 . . . . . . . . . . . . 13 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑦𝐵)))
106, 9mpbid 232 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) → (𝑥𝐴𝑦𝐵))
1110imp 406 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
12 rspe 3247 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
134, 11, 12syl2anc 584 . . . . . . . . . 10 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → ∃𝑥𝐴 𝑦𝐵)
14 abid 2716 . . . . . . . . . 10 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ ∃𝑥𝐴 𝑦𝐵)
1513, 14sylibr 234 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
16 eleq1 2827 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}))
1716ad2antrr 726 . . . . . . . . 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 4998 . . . . . . . 8 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
2018, 19eleqtrrdi 2850 . . . . . . 7 (((𝑥 = 𝑦 ∧ ∀𝑥𝐴 𝑥𝐵) ∧ 𝑥𝐴) → 𝑥 𝑥𝐴 𝐵)
2120expl 457 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2221equcoms 2017 . . . . 5 (𝑦 = 𝑥 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2322vtocleg 3553 . . . 4 (𝑥𝐴 → ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵))
2423anabsi7 671 . . 3 ((∀𝑥𝐴 𝑥𝐵𝑥𝐴) → 𝑥 𝑥𝐴 𝐵)
2524ex 412 . 2 (∀𝑥𝐴 𝑥𝐵 → (𝑥𝐴𝑥 𝑥𝐴 𝐵))
261, 2, 3, 25ssrd 4000 1 (∀𝑥𝐴 𝑥𝐵𝐴 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-iun 4998
This theorem is referenced by:  pibt2  37400
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