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Theorem uniiunlem 3001
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 2024 . . . . . 6 (y = z → (y = Bz = B))
21rexbidv 2301 . . . . 5 (y = z → (x A y = Bx A z = B))
32cbvabv 2139 . . . 4 {yx A y = B} = {zx A z = B}
43sseq1i 2942 . . 3 ({yx A y = B} ⊆ 𝐶 ↔ {zx A z = B} ⊆ 𝐶)
5 r19.23v 2399 . . . . 5 (x A (z = Bz 𝐶) ↔ (x A z = Bz 𝐶))
65albii 1335 . . . 4 (zx A (z = Bz 𝐶) ↔ z(x A z = Bz 𝐶))
7 ralcom4 2549 . . . 4 (x A z(z = Bz 𝐶) ↔ zx A (z = Bz 𝐶))
8 abss 2982 . . . 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 1398 . . . . 5 z B 𝐶
12 eleq1 2078 . . . . 5 (z = B → (z 𝐶B 𝐶))
1311, 12ceqsalg 2555 . . . 4 (B 𝐷 → (z(z = Bz 𝐶) ↔ B 𝐶))
1413ralimi 2358 . . 3 (x A B 𝐷x A (z(z = Bz 𝐶) ↔ B 𝐶))
15 ralbi 2419 . . 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 1224   = wceq 1226   wcel 1370  {cab 2004  wral 2280  wrex 2281  wss 2890
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-in 2897  df-ss 2904
This theorem is referenced by: (None)
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