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Mirrors > Home > ILE Home > Th. List > elxp6 | GIF version |
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5026. (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 | elex 2697 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V) | |
2 | opexg 4150 | . . . 4 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V) | |
3 | 2 | adantl 275 | . . 3 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V) |
4 | eleq1 2202 | . . . 4 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → (𝐴 ∈ V ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V)) | |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V)) |
6 | 3, 5 | mpbird 166 | . 2 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V) |
7 | 1stvalg 6040 | . . . . . 6 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) | |
8 | 2ndvalg 6041 | . . . . . 6 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) | |
9 | 7, 8 | opeq12d 3713 | . . . . 5 ⊢ (𝐴 ∈ V → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉) |
10 | 9 | eqeq2d 2151 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ↔ 𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉)) |
11 | 7 | eleq1d 2208 | . . . . 5 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵)) |
12 | 8 | eleq1d 2208 | . . . . 5 ⊢ (𝐴 ∈ V → ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶)) |
13 | 11, 12 | anbi12d 464 | . . . 4 ⊢ (𝐴 ∈ V → (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) |
14 | 10, 13 | anbi12d 464 | . . 3 ⊢ (𝐴 ∈ V → ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))) |
15 | elxp4 5026 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | |
16 | 14, 15 | syl6rbbr 198 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))) |
17 | 1, 6, 16 | pm5.21nii 693 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 {csn 3527 〈cop 3530 ∪ cuni 3736 × cxp 4537 dom cdm 4539 ran crn 4540 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: elxp7 6068 oprssdmm 6069 eqopi 6070 1st2nd2 6073 eldju2ndl 6957 eldju2ndr 6958 qredeu 11778 qnumdencl 11865 tx1cn 12438 tx2cn 12439 psmetxrge0 12501 xmetxpbl 12677 |
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