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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 5150 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) | |
2 | relxp 5695 | . . . . . 6 ⊢ Rel ({𝑥} × 𝐵) | |
3 | 2 | rgenw 3066 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵) |
4 | reliun 5817 | . . . . 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 5733 | . . 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 3493 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
9 | 8 | adantr 482 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V) |
10 | nfiu1 5032 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
11 | 10 | nfel2 2922 | . . . 4 ⊢ Ⅎ𝑥⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
12 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) | |
13 | 11, 12 | nfbi 1907 | . . 3 ⊢ Ⅎ𝑥(⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
14 | opeq1 4874 | . . . . 5 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩) | |
15 | 14 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
16 | eleq1 2822 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
17 | opeliunxp2.1 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) | |
18 | 17 | eleq2d 2820 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐷 ∈ 𝐵 ↔ 𝐷 ∈ 𝐸)) |
19 | 16, 18 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
20 | 15, 19 | bibi12d 346 | . . 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 5744 | . . 3 ⊢ (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
22 | 13, 20, 21 | vtoclg1f 3556 | . 2 ⊢ (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
23 | 7, 9, 22 | pm5.21nii 380 | 1 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 {csn 4629 ⟨cop 4635 ∪ ciun 4998 class class class wbr 5149 × cxp 5675 Rel wrel 5682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-iun 5000 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 |
This theorem is referenced by: mpoxopn0yelv 8198 mpoxopxnop0 8200 eldmcoa 18015 dmdprd 19868 ply1frcl 21837 cnextfres 23573 eldv 25415 perfdvf 25420 eltayl 25872 dfcnv2 31901 cvmliftlem1 34276 filnetlem3 35265 |
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