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Theorem uniiunlem 3180
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 2144 . . . . . 6 (𝑦 = 𝑧 → (𝑦 = 𝐵𝑧 = 𝐵))
21rexbidv 2436 . . . . 5 (𝑦 = 𝑧 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32cbvabv 2262 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
43sseq1i 3118 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶)
5 r19.23v 2539 . . . . 5 (∀𝑥𝐴 (𝑧 = 𝐵𝑧𝐶) ↔ (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
65albii 1446 . . . 4 (∀𝑧𝑥𝐴 (𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑧(∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
7 ralcom4 2703 . . . 4 (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑧𝑥𝐴 (𝑧 = 𝐵𝑧𝐶))
8 abss 3161 . . . 4 ({𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶 ↔ ∀𝑧(∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
96, 7, 83bitr4i 211 . . 3 (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶)
104, 9bitr4i 186 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ ∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶))
11 nfv 1508 . . . . 5 𝑧 𝐵𝐶
12 eleq1 2200 . . . . 5 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
1311, 12ceqsalg 2709 . . . 4 (𝐵𝐷 → (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶))
1413ralimi 2493 . . 3 (∀𝑥𝐴 𝐵𝐷 → ∀𝑥𝐴 (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶))
15 ralbi 2562 . . 3 (∀𝑥𝐴 (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶) → (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑥𝐴 𝐵𝐶))
1614, 15syl 14 . 2 (∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑥𝐴 𝐵𝐶))
1710, 16syl5rbb 192 1 (∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329   = wceq 1331  wcel 1480  {cab 2123  wral 2414  wrex 2415  wss 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079
This theorem is referenced by:  iunopn  12158
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