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Theorem opeliunxp2f 7865
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5702. (Contributed by AV, 25-Oct-2020.)
Hypotheses
Ref Expression
opeliunxp2f.f 𝑥𝐸
opeliunxp2f.e (𝑥 = 𝐶𝐵 = 𝐸)
Assertion
Ref Expression
opeliunxp2f (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐸(𝑥)

Proof of Theorem opeliunxp2f
StepHypRef Expression
1 df-br 5058 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
2 relxp 5566 . . . . . 6 Rel ({𝑥} × 𝐵)
32rgenw 3147 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
4 reliun 5682 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
53, 4mpbir 232 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
65brrelex1i 5601 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷𝐶 ∈ V)
71, 6sylbir 236 . 2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8 elex 3510 . . 3 (𝐶𝐴𝐶 ∈ V)
98adantr 481 . 2 ((𝐶𝐴𝐷𝐸) → 𝐶 ∈ V)
10 nfiu1 4944 . . . . 5 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
1110nfel2 2993 . . . 4 𝑥𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)
12 nfv 1906 . . . . 5 𝑥 𝐶𝐴
13 opeliunxp2f.f . . . . . 6 𝑥𝐸
1413nfel2 2993 . . . . 5 𝑥 𝐷𝐸
1512, 14nfan 1891 . . . 4 𝑥(𝐶𝐴𝐷𝐸)
1611, 15nfbi 1895 . . 3 𝑥(⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
17 opeq1 4795 . . . . 5 (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1817eleq1d 2894 . . . 4 (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
19 eleq1 2897 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
20 opeliunxp2f.e . . . . . 6 (𝑥 = 𝐶𝐵 = 𝐸)
2120eleq2d 2895 . . . . 5 (𝑥 = 𝐶 → (𝐷𝐵𝐷𝐸))
2219, 21anbi12d 630 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴𝐷𝐵) ↔ (𝐶𝐴𝐷𝐸)))
2318, 22bibi12d 347 . . 3 (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))))
24 opeliunxp 5612 . . 3 (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵))
2516, 23, 24vtoclg1f 3564 . 2 (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸)))
267, 9, 25pm5.21nii 380 1 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wnfc 2958  wral 3135  Vcvv 3492  {csn 4557  cop 4563   ciun 4910   class class class wbr 5057   × cxp 5546  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-iun 4912  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  mpoxeldm  7866  fsumcom2  15117  fprodcom2  15326
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