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Theorem opeliunxp2f 6101
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 4647. (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 3898 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
2 relxp 4616 . . . . . 6 Rel ({𝑥} × 𝐵)
32rgenw 2462 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
4 reliun 4628 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
53, 4mpbir 145 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
65brrelex1i 4550 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷𝐶 ∈ V)
71, 6sylbir 134 . 2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8 elex 2669 . . 3 (𝐶𝐴𝐶 ∈ V)
98adantr 272 . 2 ((𝐶𝐴𝐷𝐸) → 𝐶 ∈ V)
10 nfiu1 3811 . . . . 5 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
1110nfel2 2269 . . . 4 𝑥𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)
12 nfv 1491 . . . . 5 𝑥 𝐶𝐴
13 opeliunxp2f.f . . . . . 6 𝑥𝐸
1413nfel2 2269 . . . . 5 𝑥 𝐷𝐸
1512, 14nfan 1527 . . . 4 𝑥(𝐶𝐴𝐷𝐸)
1611, 15nfbi 1551 . . 3 𝑥(⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
17 opeq1 3673 . . . . 5 (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1817eleq1d 2184 . . . 4 (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
19 eleq1 2178 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
20 opeliunxp2f.e . . . . . 6 (𝑥 = 𝐶𝐵 = 𝐸)
2120eleq2d 2185 . . . . 5 (𝑥 = 𝐶 → (𝐷𝐵𝐷𝐸))
2219, 21anbi12d 462 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴𝐷𝐵) ↔ (𝐶𝐴𝐷𝐸)))
2318, 22bibi12d 234 . . 3 (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))))
24 opeliunxp 4562 . . 3 (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵))
2516, 23, 24vtoclg1f 2717 . 2 (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸)))
267, 9, 25pm5.21nii 676 1 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  wnfc 2243  wral 2391  Vcvv 2658  {csn 3495  cop 3498   ciun 3781   class class class wbr 3897   × cxp 4505  Rel wrel 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-iun 3783  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514
This theorem is referenced by:  fisumcom2  11158
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