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Mirrors > Home > ILE Home > Th. List > dmxpm | GIF version |
Description: The domain of a cross 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.) |
Ref | Expression |
---|---|
dmxpm | ⊢ (∃x x ∈ B → dom (A × B) = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2097 | . . 3 ⊢ (x = z → (x ∈ B ↔ z ∈ B)) | |
2 | 1 | cbvexv 1792 | . 2 ⊢ (∃x x ∈ B ↔ ∃z z ∈ B) |
3 | df-xp 4294 | . . . 4 ⊢ (A × B) = {〈y, z〉 ∣ (y ∈ A ∧ z ∈ B)} | |
4 | 3 | dmeqi 4479 | . . 3 ⊢ dom (A × B) = dom {〈y, z〉 ∣ (y ∈ A ∧ z ∈ B)} |
5 | id 19 | . . . . 5 ⊢ (∃z z ∈ B → ∃z z ∈ B) | |
6 | 5 | ralrimivw 2387 | . . . 4 ⊢ (∃z z ∈ B → ∀y ∈ A ∃z z ∈ B) |
7 | dmopab3 4491 | . . . 4 ⊢ (∀y ∈ A ∃z z ∈ B ↔ dom {〈y, z〉 ∣ (y ∈ A ∧ z ∈ B)} = A) | |
8 | 6, 7 | sylib 127 | . . 3 ⊢ (∃z z ∈ B → dom {〈y, z〉 ∣ (y ∈ A ∧ z ∈ B)} = A) |
9 | 4, 8 | syl5eq 2081 | . 2 ⊢ (∃z z ∈ B → dom (A × B) = A) |
10 | 2, 9 | sylbi 114 | 1 ⊢ (∃x x ∈ B → dom (A × B) = A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∀wral 2300 {copab 3808 × cxp 4286 dom cdm 4288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-dm 4298 |
This theorem is referenced by: dmxpinm 4499 xpid11m 4500 rnxpm 4695 ssxpbm 4699 ssxp1 4700 xpexr2m 4705 relrelss 4787 unixpm 4796 xpiderm 6113 |
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