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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 5144 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ 〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) | |
| 2 | relxp 5703 | . . . . . 6 ⊢ Rel ({𝑥} × 𝐵) | |
| 3 | 2 | rgenw 3065 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵) |
| 4 | reliun 5826 | . . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)) | |
| 5 | 3, 4 | mpbir 231 | . . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
| 6 | 5 | brrelex1i 5741 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 → 𝐶 ∈ V) |
| 7 | 1, 6 | sylbir 235 | . 2 ⊢ (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V) |
| 8 | elex 3501 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
| 9 | 8 | adantr 480 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V) |
| 10 | nfiu1 5027 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
| 11 | 10 | nfel2 2924 | . . . 4 ⊢ Ⅎ𝑥〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
| 12 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) | |
| 13 | 11, 12 | nfbi 1903 | . . 3 ⊢ Ⅎ𝑥(〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
| 14 | opeq1 4873 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝐷〉 = 〈𝐶, 𝐷〉) | |
| 15 | 14 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
| 16 | eleq1 2829 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
| 17 | opeliunxp2.1 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) | |
| 18 | 17 | eleq2d 2827 | . . . . 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 345 | . . 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 5752 | . . 3 ⊢ (〈𝑥, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
| 22 | 13, 20, 21 | vtoclg1f 3570 | . 2 ⊢ (𝐶 ∈ V → (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
| 23 | 7, 9, 22 | pm5.21nii 378 | 1 ⊢ (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 {csn 4626 〈cop 4632 ∪ ciun 4991 class class class wbr 5143 × cxp 5683 Rel wrel 5690 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: mpoxopn0yelv 8238 mpoxopxnop0 8240 eldmcoa 18110 dmdprd 20018 ply1frcl 22322 cnextfres 24077 eldv 25933 perfdvf 25938 eltayl 26401 dfcnv2 32686 gsumwrd2dccat 33070 cvmliftlem1 35290 filnetlem3 36381 |
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