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Mirrors > Home > MPE Home > Th. List > elxp6 | Structured version Visualization version GIF version |
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7003. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
elxp6 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp4 7003 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | |
2 | 1stval 7061 | . . . . 5 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} | |
3 | 2ndval 7062 | . . . . 5 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} | |
4 | 2, 3 | opeq12i 4345 | . . . 4 ⊢ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 |
5 | 4 | eqeq2i 2622 | . . 3 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ↔ 𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉) |
6 | 2 | eleq1i 2679 | . . . 4 ⊢ ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵) |
7 | 3 | eleq1i 2679 | . . . 4 ⊢ ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶) |
8 | 6, 7 | anbi12i 729 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) |
9 | 5, 8 | anbi12i 729 | . 2 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) |
10 | 1, 9 | bitr4i 266 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 ∪ cuni 4372 × cxp 5036 dom cdm 5038 ran crn 5039 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: elxp7 7092 eqopi 7093 1st2nd2 7096 r0weon 8718 qredeu 15210 qnumdencl 15285 tx1cn 21222 tx2cn 21223 txhaus 21260 psmetxrge0 21928 xppreima 28829 ofpreima2 28849 smatrcl 29190 1stmbfm 29649 2ndmbfm 29650 oddpwdcv 29744 |
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