ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeliunxp2 GIF version

Theorem opeliunxp2 4782
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⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
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 4019 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
2 relxp 4750 . . . . . 6 Rel ({𝑥} × 𝐵)
32rgenw 2545 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
4 reliun 4762 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
53, 4mpbir 146 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
65brrelex1i 4684 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷𝐶 ∈ V)
71, 6sylbir 135 . 2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8 elex 2763 . . 3 (𝐶𝐴𝐶 ∈ V)
98adantr 276 . 2 ((𝐶𝐴𝐷𝐸) → 𝐶 ∈ V)
10 nfcv 2332 . . 3 𝑥𝐶
11 nfiu1 3931 . . . . 5 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
1211nfel2 2345 . . . 4 𝑥𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)
13 nfv 1539 . . . 4 𝑥(𝐶𝐴𝐷𝐸)
1412, 13nfbi 1600 . . 3 𝑥(⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
15 opeq1 3793 . . . . 5 (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1615eleq1d 2258 . . . 4 (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
17 eleq1 2252 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
18 opeliunxp2.1 . . . . . 6 (𝑥 = 𝐶𝐵 = 𝐸)
1918eleq2d 2259 . . . . 5 (𝑥 = 𝐶 → (𝐷𝐵𝐷𝐸))
2017, 19anbi12d 473 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴𝐷𝐵) ↔ (𝐶𝐴𝐷𝐸)))
2116, 20bibi12d 235 . . 3 (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))))
22 opeliunxp 4696 . . 3 (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵))
2310, 14, 21, 22vtoclgf 2810 . 2 (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸)))
247, 9, 23pm5.21nii 705 1 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752  {csn 3607  cop 3610   ciun 3901   class class class wbr 4018   × cxp 4639  Rel wrel 4646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-iun 3903  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648
This theorem is referenced by:  mpoxopn0yelv  6258  eldvap  14554
  Copyright terms: Public domain W3C validator