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Theorem opeliunxp2f 8225
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5845. (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 5154 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
2 relxp 5700 . . . . . 6 Rel ({𝑥} × 𝐵)
32rgenw 3055 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
4 reliun 5822 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
53, 4mpbir 230 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
65brrelex1i 5738 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷𝐶 ∈ V)
71, 6sylbir 234 . 2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8 elex 3482 . . 3 (𝐶𝐴𝐶 ∈ V)
98adantr 479 . 2 ((𝐶𝐴𝐷𝐸) → 𝐶 ∈ V)
10 nfiu1 5035 . . . . 5 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
1110nfel2 2911 . . . 4 𝑥𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)
12 nfv 1910 . . . . 5 𝑥 𝐶𝐴
13 opeliunxp2f.f . . . . . 6 𝑥𝐸
1413nfel2 2911 . . . . 5 𝑥 𝐷𝐸
1512, 14nfan 1895 . . . 4 𝑥(𝐶𝐴𝐷𝐸)
1611, 15nfbi 1899 . . 3 𝑥(⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
17 opeq1 4879 . . . . 5 (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1817eleq1d 2811 . . . 4 (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
19 eleq1 2814 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
20 opeliunxp2f.e . . . . . 6 (𝑥 = 𝐶𝐵 = 𝐸)
2120eleq2d 2812 . . . . 5 (𝑥 = 𝐶 → (𝐷𝐵𝐷𝐸))
2219, 21anbi12d 630 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴𝐷𝐵) ↔ (𝐶𝐴𝐷𝐸)))
2318, 22bibi12d 344 . . 3 (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))))
24 opeliunxp 5749 . . 3 (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵))
2516, 23, 24vtoclg1f 3551 . 2 (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸)))
267, 9, 25pm5.21nii 377 1 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wnfc 2876  wral 3051  Vcvv 3462  {csn 4633  cop 4639   ciun 5001   class class class wbr 5153   × cxp 5680  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-iun 5003  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689
This theorem is referenced by:  mpoxeldm  8226  fsumcom2  15778  fprodcom2  15986  iunsnima2  32539
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