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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 5031 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ 〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) | |
2 | relxp 5537 | . . . . . 6 ⊢ Rel ({𝑥} × 𝐵) | |
3 | 2 | rgenw 3118 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵) |
4 | reliun 5653 | . . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)) | |
5 | 3, 4 | mpbir 234 | . . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
6 | 5 | brrelex1i 5572 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 → 𝐶 ∈ V) |
7 | 1, 6 | sylbir 238 | . 2 ⊢ (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V) |
8 | elex 3459 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
9 | 8 | adantr 484 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V) |
10 | nfiu1 4915 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
11 | 10 | nfel2 2973 | . . . 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 4763 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝐷〉 = 〈𝐶, 𝐷〉) | |
15 | 14 | eleq1d 2874 | . . . 4 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
16 | eleq1 2877 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
17 | opeliunxp2.1 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) | |
18 | 17 | eleq2d 2875 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐷 ∈ 𝐵 ↔ 𝐷 ∈ 𝐸)) |
19 | 16, 18 | anbi12d 633 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
20 | 15, 19 | bibi12d 349 | . . 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 5583 | . . 3 ⊢ (〈𝑥, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
22 | 13, 20, 21 | vtoclg1f 3514 | . 2 ⊢ (𝐶 ∈ V → (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
23 | 7, 9, 22 | pm5.21nii 383 | 1 ⊢ (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 {csn 4525 〈cop 4531 ∪ ciun 4881 class class class wbr 5030 × cxp 5517 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-iun 4883 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: mpoxopn0yelv 7862 mpoxopxnop0 7864 eldmcoa 17317 dmdprd 19113 ply1frcl 20942 cnextfres 22674 eldv 24501 perfdvf 24506 eltayl 24955 dfcnv2 30439 cvmliftlem1 32645 filnetlem3 33841 |
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