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Theorem uniiunlem 3092
Description: A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Assertion
Ref Expression
uniiunlem (∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem uniiunlem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2089 . . . . . 6 (𝑦 = 𝑧 → (𝑦 = 𝐵𝑧 = 𝐵))
21rexbidv 2375 . . . . 5 (𝑦 = 𝑧 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32cbvabv 2206 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
43sseq1i 3033 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶)
5 r19.23v 2475 . . . . 5 (∀𝑥𝐴 (𝑧 = 𝐵𝑧𝐶) ↔ (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
65albii 1400 . . . 4 (∀𝑧𝑥𝐴 (𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑧(∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
7 ralcom4 2631 . . . 4 (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑧𝑥𝐴 (𝑧 = 𝐵𝑧𝐶))
8 abss 3073 . . . 4 ({𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶 ↔ ∀𝑧(∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
96, 7, 83bitr4i 210 . . 3 (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶)
104, 9bitr4i 185 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ ∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶))
11 nfv 1462 . . . . 5 𝑧 𝐵𝐶
12 eleq1 2145 . . . . 5 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
1311, 12ceqsalg 2637 . . . 4 (𝐵𝐷 → (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶))
1413ralimi 2432 . . 3 (∀𝑥𝐴 𝐵𝐷 → ∀𝑥𝐴 (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶))
15 ralbi 2495 . . 3 (∀𝑥𝐴 (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶) → (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑥𝐴 𝐵𝐶))
1614, 15syl 14 . 2 (∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑥𝐴 𝐵𝐶))
1710, 16syl5rbb 191 1 (∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  wss 2983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-in 2989  df-ss 2996
This theorem is referenced by: (None)
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