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Mirrors > Home > MPE Home > Th. List > Mathboxes > elxp8 | Structured version Visualization version GIF version |
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7866. (Contributed by ML, 19-Oct-2020.) |
Ref | Expression |
---|---|
elxp8 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp1st 7863 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) | |
2 | ssv 3945 | . . . . 5 ⊢ 𝐵 ⊆ V | |
3 | ssid 3943 | . . . . 5 ⊢ 𝐶 ⊆ 𝐶 | |
4 | xpss12 5604 | . . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶)) | |
5 | 2, 3, 4 | mp2an 689 | . . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶) |
6 | 5 | sseli 3917 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶)) |
7 | 1, 6 | jca 512 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
8 | xpss 5605 | . . . . 5 ⊢ (V × 𝐶) ⊆ (V × V) | |
9 | 8 | sseli 3917 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V)) |
10 | 9 | adantl 482 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V)) |
11 | xp2nd 7864 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) | |
12 | 11 | anim2i 617 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) |
13 | elxp7 7866 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
14 | 10, 12, 13 | sylanbrc 583 | . 2 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶)) |
15 | 7, 14 | impbii 208 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 × cxp 5587 ‘cfv 6433 1st c1st 7829 2nd c2nd 7830 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fv 6441 df-1st 7831 df-2nd 7832 |
This theorem is referenced by: finxpsuclem 35568 |
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