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| Description: The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2140, ax-11 2156, ax-12 2176. (Revised by SN, 12-Aug-2025.) | 
| Ref | Expression | 
|---|---|
| dmxp | ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3483 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | eldm 5910 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦 𝑥(𝐴 × 𝐵)𝑦) | 
| 3 | brxp 5733 | . . . . 5 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
| 4 | 3 | exbii 1847 | . . . 4 ⊢ (∃𝑦 𝑥(𝐴 × 𝐵)𝑦 ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | 
| 5 | 19.42v 1952 | . . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)) | |
| 6 | 2, 4, 5 | 3bitri 297 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)) | 
| 7 | n0 4352 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵) | |
| 8 | 7 | biimpi 216 | . . . 4 ⊢ (𝐵 ≠ ∅ → ∃𝑦 𝑦 ∈ 𝐵) | 
| 9 | 8 | biantrud 531 | . . 3 ⊢ (𝐵 ≠ ∅ → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵))) | 
| 10 | 6, 9 | bitr4id 290 | . 2 ⊢ (𝐵 ≠ ∅ → (𝑥 ∈ dom (𝐴 × 𝐵) ↔ 𝑥 ∈ 𝐴)) | 
| 11 | 10 | eqrdv 2734 | 1 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 class class class wbr 5142 × cxp 5682 dom cdm 5684 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-dm 5694 | 
| This theorem is referenced by: dmxpid 5940 rnxp 6189 dmxpss 6190 ssxpb 6193 relrelss 6292 unixp 6301 xpexr2 7942 xpexcnv 7943 frxp 8152 mpocurryd 8295 fodomr 9169 fodomfir 9369 nqerf 10971 dmtrclfv 15058 pwsbas 17533 pwsle 17538 imasaddfnlem 17574 imasvscafn 17583 efgrcl 19734 frlmip 21799 txindislem 23642 metustexhalf 24570 rrxip 25425 dveq0 26040 dv11cn 26041 noxp1o 27709 noextendseq 27713 bdayfo 27723 noetasuplem2 27780 noetasuplem4 27782 noetainflem2 27784 noetainflem4 27786 dmdju 32658 mbfmcst 34262 eulerpartlemt 34374 0rrv 34454 curf 37606 curunc 37610 ismgmOLD 37858 diophrw 42775 onnog 43447 onnobdayg 43448 bdaybndbday 43450 | 
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