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Theorem opeliunxp2 4403
 Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
opeliunxp2.1 (x = 𝐶B = 𝐸)
Assertion
Ref Expression
opeliunxp2 (⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸))
Distinct variable groups:   x,𝐶   x,𝐷   x,𝐸   x,A
Allowed substitution hint:   B(x)

Proof of Theorem opeliunxp2
StepHypRef Expression
1 df-br 3739 . . 3 (𝐶 x A ({x} × B)𝐷 ↔ ⟨𝐶, 𝐷 x A ({x} × B))
2 relxp 4374 . . . . . 6 Rel ({x} × B)
32rgenw 2354 . . . . 5 x A Rel ({x} × B)
4 reliun 4385 . . . . 5 (Rel x A ({x} × B) ↔ x A Rel ({x} × B))
53, 4mpbir 134 . . . 4 Rel x A ({x} × B)
65brrelexi 4311 . . 3 (𝐶 x A ({x} × B)𝐷𝐶 V)
71, 6sylbir 125 . 2 (⟨𝐶, 𝐷 x A ({x} × B) → 𝐶 V)
8 elex 2543 . . 3 (𝐶 A𝐶 V)
98adantr 261 . 2 ((𝐶 A 𝐷 𝐸) → 𝐶 V)
10 nfcv 2160 . . 3 x𝐶
11 nfiu1 3661 . . . . 5 x x A ({x} × B)
1211nfel2 2172 . . . 4 x𝐶, 𝐷 x A ({x} × B)
13 nfv 1402 . . . 4 x(𝐶 A 𝐷 𝐸)
1412, 13nfbi 1463 . . 3 x(⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸))
15 opeq1 3523 . . . . 5 (x = 𝐶 → ⟨x, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1615eleq1d 2088 . . . 4 (x = 𝐶 → (⟨x, 𝐷 x A ({x} × B) ↔ ⟨𝐶, 𝐷 x A ({x} × B)))
17 eleq1 2082 . . . . 5 (x = 𝐶 → (x A𝐶 A))
18 opeliunxp2.1 . . . . . 6 (x = 𝐶B = 𝐸)
1918eleq2d 2089 . . . . 5 (x = 𝐶 → (𝐷 B𝐷 𝐸))
2017, 19anbi12d 445 . . . 4 (x = 𝐶 → ((x A 𝐷 B) ↔ (𝐶 A 𝐷 𝐸)))
2116, 20bibi12d 224 . . 3 (x = 𝐶 → ((⟨x, 𝐷 x A ({x} × B) ↔ (x A 𝐷 B)) ↔ (⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸))))
22 opeliunxp 4322 . . 3 (⟨x, 𝐷 x A ({x} × B) ↔ (x A 𝐷 B))
2310, 14, 21, 22vtoclgf 2589 . 2 (𝐶 V → (⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸)))
247, 9, 23pm5.21nii 607 1 (⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∀wral 2284  Vcvv 2535  {csn 3350  ⟨cop 3353  ∪ ciun 3631   class class class wbr 3738   × cxp 4270  Rel wrel 4277 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279 This theorem is referenced by:  mpt2xopn0yelv  5776
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