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Theorem opeliunxp2 4418
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 3755 . . 3 (𝐶 x A ({x} × B)𝐷 ↔ ⟨𝐶, 𝐷 x A ({x} × B))
2 relxp 4389 . . . . . 6 Rel ({x} × B)
32rgenw 2370 . . . . 5 x A Rel ({x} × B)
4 reliun 4400 . . . . 5 (Rel x A ({x} × B) ↔ x A Rel ({x} × B))
53, 4mpbir 134 . . . 4 Rel x A ({x} × B)
65brrelexi 4326 . . 3 (𝐶 x A ({x} × B)𝐷𝐶 V)
71, 6sylbir 125 . 2 (⟨𝐶, 𝐷 x A ({x} × B) → 𝐶 V)
8 elex 2560 . . 3 (𝐶 A𝐶 V)
98adantr 261 . 2 ((𝐶 A 𝐷 𝐸) → 𝐶 V)
10 nfcv 2175 . . 3 x𝐶
11 nfiu1 3677 . . . . 5 x x A ({x} × B)
1211nfel2 2187 . . . 4 x𝐶, 𝐷 x A ({x} × B)
13 nfv 1418 . . . 4 x(𝐶 A 𝐷 𝐸)
1412, 13nfbi 1478 . . 3 x(⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸))
15 opeq1 3539 . . . . 5 (x = 𝐶 → ⟨x, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1615eleq1d 2103 . . . 4 (x = 𝐶 → (⟨x, 𝐷 x A ({x} × B) ↔ ⟨𝐶, 𝐷 x A ({x} × B)))
17 eleq1 2097 . . . . 5 (x = 𝐶 → (x A𝐶 A))
18 opeliunxp2.1 . . . . . 6 (x = 𝐶B = 𝐸)
1918eleq2d 2104 . . . . 5 (x = 𝐶 → (𝐷 B𝐷 𝐸))
2017, 19anbi12d 442 . . . 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 4337 . . 3 (⟨x, 𝐷 x A ({x} × B) ↔ (x A 𝐷 B))
2310, 14, 21, 22vtoclgf 2606 . 2 (𝐶 V → (⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸)))
247, 9, 23pm5.21nii 619 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 1242   wcel 1390  wral 2300  Vcvv 2551  {csn 3366  cop 3369   ciun 3647   class class class wbr 3754   × cxp 4285  Rel wrel 4292
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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  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-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-iun 3649  df-br 3755  df-opab 3809  df-xp 4293  df-rel 4294
This theorem is referenced by:  mpt2xopn0yelv  5792
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