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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 7468. (Contributed by ML, 19-Oct-2020.) |
Ref | Expression |
---|---|
elxp8 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp1st 7465 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) | |
2 | ssv 3850 | . . . . 5 ⊢ 𝐵 ⊆ V | |
3 | ssid 3848 | . . . . 5 ⊢ 𝐶 ⊆ 𝐶 | |
4 | xpss12 5361 | . . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶)) | |
5 | 2, 3, 4 | mp2an 683 | . . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶) |
6 | 5 | sseli 3823 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶)) |
7 | 1, 6 | jca 507 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
8 | xpss 5362 | . . . . 5 ⊢ (V × 𝐶) ⊆ (V × V) | |
9 | 8 | sseli 3823 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V)) |
10 | 9 | adantl 475 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V)) |
11 | xp2nd 7466 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) | |
12 | 11 | anim2i 610 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) |
13 | elxp7 7468 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
14 | 10, 12, 13 | sylanbrc 578 | . 2 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶)) |
15 | 7, 14 | impbii 201 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2164 Vcvv 3414 ⊆ wss 3798 × cxp 5344 ‘cfv 6127 1st c1st 7431 2nd c2nd 7432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fv 6135 df-1st 7433 df-2nd 7434 |
This theorem is referenced by: finxpsuclem 33774 |
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