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Mirrors > Home > MPE Home > Th. List > opeliunxp2 | Structured version Visualization version GIF version |
Description: Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
opeliunxp2.1 | ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) |
Ref | Expression |
---|---|
opeliunxp2 | ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5148 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) | |
2 | relxp 5693 | . . . . . 6 ⊢ Rel ({𝑥} × 𝐵) | |
3 | 2 | rgenw 3063 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵) |
4 | reliun 5815 | . . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)) | |
5 | 3, 4 | mpbir 230 | . . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
6 | 5 | brrelex1i 5731 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 → 𝐶 ∈ V) |
7 | 1, 6 | sylbir 234 | . 2 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V) |
8 | elex 3491 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
9 | 8 | adantr 479 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V) |
10 | nfiu1 5030 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
11 | 10 | nfel2 2919 | . . . 4 ⊢ Ⅎ𝑥⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
12 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) | |
13 | 11, 12 | nfbi 1904 | . . 3 ⊢ Ⅎ𝑥(⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
14 | opeq1 4872 | . . . . 5 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩) | |
15 | 14 | eleq1d 2816 | . . . 4 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
16 | eleq1 2819 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
17 | opeliunxp2.1 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) | |
18 | 17 | eleq2d 2817 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐷 ∈ 𝐵 ↔ 𝐷 ∈ 𝐸)) |
19 | 16, 18 | anbi12d 629 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
20 | 15, 19 | bibi12d 344 | . . 3 ⊢ (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)))) |
21 | opeliunxp 5742 | . . 3 ⊢ (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
22 | 13, 20, 21 | vtoclg1f 3557 | . 2 ⊢ (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
23 | 7, 9, 22 | pm5.21nii 377 | 1 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 Vcvv 3472 {csn 4627 ⟨cop 4633 ∪ ciun 4996 class class class wbr 5147 × cxp 5673 Rel wrel 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-iun 4998 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 |
This theorem is referenced by: mpoxopn0yelv 8200 mpoxopxnop0 8202 eldmcoa 18019 dmdprd 19909 ply1frcl 22057 cnextfres 23793 eldv 25647 perfdvf 25652 eltayl 26108 dfcnv2 32168 cvmliftlem1 34574 filnetlem3 35568 |
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