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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 8022. (Contributed by ML, 19-Oct-2020.) |
Ref | Expression |
---|---|
elxp8 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp1st 8019 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) | |
2 | ssv 4002 | . . . . 5 ⊢ 𝐵 ⊆ V | |
3 | ssid 4000 | . . . . 5 ⊢ 𝐶 ⊆ 𝐶 | |
4 | xpss12 5687 | . . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶)) | |
5 | 2, 3, 4 | mp2an 691 | . . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶) |
6 | 5 | sseli 3974 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶)) |
7 | 1, 6 | jca 511 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
8 | xpss 5688 | . . . . 5 ⊢ (V × 𝐶) ⊆ (V × V) | |
9 | 8 | sseli 3974 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V)) |
10 | 9 | adantl 481 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V)) |
11 | xp2nd 8020 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) | |
12 | 11 | anim2i 616 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) |
13 | elxp7 8022 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
14 | 10, 12, 13 | sylanbrc 582 | . 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 395 ∈ wcel 2099 Vcvv 3469 ⊆ wss 3944 × cxp 5670 ‘cfv 6542 1st c1st 7985 2nd c2nd 7986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fv 6550 df-1st 7987 df-2nd 7988 |
This theorem is referenced by: finxpsuclem 36812 |
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