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Theorem uniiunlem 3022
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 (x A B 𝐷 → (x A B 𝐶 ↔ {yx A 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 2043 . . . . . 6 (y = z → (y = Bz = B))
21rexbidv 2321 . . . . 5 (y = z → (x A y = Bx A z = B))
32cbvabv 2158 . . . 4 {yx A y = B} = {zx A z = B}
43sseq1i 2963 . . 3 ({yx A y = B} ⊆ 𝐶 ↔ {zx A z = B} ⊆ 𝐶)
5 r19.23v 2419 . . . . 5 (x A (z = Bz 𝐶) ↔ (x A z = Bz 𝐶))
65albii 1356 . . . 4 (zx A (z = Bz 𝐶) ↔ z(x A z = Bz 𝐶))
7 ralcom4 2570 . . . 4 (x A z(z = Bz 𝐶) ↔ zx A (z = Bz 𝐶))
8 abss 3003 . . . 4 ({zx A z = B} ⊆ 𝐶z(x A z = Bz 𝐶))
96, 7, 83bitr4i 201 . . 3 (x A z(z = Bz 𝐶) ↔ {zx A z = B} ⊆ 𝐶)
104, 9bitr4i 176 . 2 ({yx A y = B} ⊆ 𝐶x A z(z = Bz 𝐶))
11 nfv 1418 . . . . 5 z B 𝐶
12 eleq1 2097 . . . . 5 (z = B → (z 𝐶B 𝐶))
1311, 12ceqsalg 2576 . . . 4 (B 𝐷 → (z(z = Bz 𝐶) ↔ B 𝐶))
1413ralimi 2378 . . 3 (x A B 𝐷x A (z(z = Bz 𝐶) ↔ B 𝐶))
15 ralbi 2439 . . 3 (x A (z(z = Bz 𝐶) ↔ B 𝐶) → (x A z(z = Bz 𝐶) ↔ x A B 𝐶))
1614, 15syl 14 . 2 (x A B 𝐷 → (x A z(z = Bz 𝐶) ↔ x A B 𝐶))
1710, 16syl5rbb 182 1 (x A B 𝐷 → (x A B 𝐶 ↔ {yx A y = B} ⊆ 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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