ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uniiunlem GIF version

Theorem uniiunlem 3055
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 2062 . . . . . 6 (𝑦 = 𝑧 → (𝑦 = 𝐵𝑧 = 𝐵))
21rexbidv 2344 . . . . 5 (𝑦 = 𝑧 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32cbvabv 2177 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
43sseq1i 2996 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶)
5 r19.23v 2442 . . . . 5 (∀𝑥𝐴 (𝑧 = 𝐵𝑧𝐶) ↔ (∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
65albii 1375 . . . 4 (∀𝑧𝑥𝐴 (𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑧(∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
7 ralcom4 2593 . . . 4 (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑧𝑥𝐴 (𝑧 = 𝐵𝑧𝐶))
8 abss 3036 . . . 4 ({𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶 ↔ ∀𝑧(∃𝑥𝐴 𝑧 = 𝐵𝑧𝐶))
96, 7, 83bitr4i 205 . . 3 (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ⊆ 𝐶)
104, 9bitr4i 180 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ ∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶))
11 nfv 1437 . . . . 5 𝑧 𝐵𝐶
12 eleq1 2116 . . . . 5 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
1311, 12ceqsalg 2599 . . . 4 (𝐵𝐷 → (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶))
1413ralimi 2401 . . 3 (∀𝑥𝐴 𝐵𝐷 → ∀𝑥𝐴 (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶))
15 ralbi 2462 . . 3 (∀𝑥𝐴 (∀𝑧(𝑧 = 𝐵𝑧𝐶) ↔ 𝐵𝐶) → (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑥𝐴 𝐵𝐶))
1614, 15syl 14 . 2 (∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴𝑧(𝑧 = 𝐵𝑧𝐶) ↔ ∀𝑥𝐴 𝐵𝐶))
1710, 16syl5rbb 186 1 (∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  wss 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator