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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 4654 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ 〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) | |
2 | relxp 5227 | . . . . . 6 ⊢ Rel ({𝑥} × 𝐵) | |
3 | 2 | rgenw 2924 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵) |
4 | reliun 5239 | . . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)) | |
5 | 3, 4 | mpbir 221 | . . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
6 | 5 | brrelexi 5158 | . . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 → 𝐶 ∈ V) |
7 | 1, 6 | sylbir 225 | . 2 ⊢ (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V) |
8 | elex 3212 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
9 | 8 | adantr 481 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V) |
10 | nfiu1 4550 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
11 | 10 | nfel2 2781 | . . . 4 ⊢ Ⅎ𝑥〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
12 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) | |
13 | 11, 12 | nfbi 1833 | . . 3 ⊢ Ⅎ𝑥(〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
14 | opeq1 4402 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝐷〉 = 〈𝐶, 𝐷〉) | |
15 | 14 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
16 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
17 | opeliunxp2.1 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) | |
18 | 17 | eleq2d 2687 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐷 ∈ 𝐵 ↔ 𝐷 ∈ 𝐸)) |
19 | 16, 18 | anbi12d 747 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
20 | 15, 19 | bibi12d 335 | . . 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 5170 | . . 3 ⊢ (〈𝑥, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
22 | 13, 20, 21 | vtoclg1f 3265 | . 2 ⊢ (𝐶 ∈ V → (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))) |
23 | 7, 9, 22 | pm5.21nii 368 | 1 ⊢ (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 {csn 4177 〈cop 4183 ∪ ciun 4520 class class class wbr 4653 × cxp 5112 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-iun 4522 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 |
This theorem is referenced by: mpt2xopn0yelv 7339 mpt2xopxnop0 7341 eldmcoa 16715 dmdprd 18397 ply1frcl 19683 cnextfres 21873 eldv 23662 perfdvf 23667 eltayl 24114 dfcnv2 29476 cvmliftlem1 31267 filnetlem3 32375 |
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