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Mirrors > Home > ILE Home > Th. List > xpexr2m | GIF version |
Description: If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.) |
Ref | Expression |
---|---|
xpexr2m | ⊢ (((A × B) ∈ 𝐶 ∧ ∃x x ∈ (A × B)) → (A ∈ V ∧ B ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpm 4688 | . 2 ⊢ ((∃𝑎 𝑎 ∈ A ∧ ∃𝑏 𝑏 ∈ B) ↔ ∃x x ∈ (A × B)) | |
2 | dmxpm 4498 | . . . . . 6 ⊢ (∃𝑏 𝑏 ∈ B → dom (A × B) = A) | |
3 | 2 | adantl 262 | . . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ B) → dom (A × B) = A) |
4 | dmexg 4539 | . . . . . 6 ⊢ ((A × B) ∈ 𝐶 → dom (A × B) ∈ V) | |
5 | 4 | adantr 261 | . . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ B) → dom (A × B) ∈ V) |
6 | 3, 5 | eqeltrrd 2112 | . . . 4 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ B) → A ∈ V) |
7 | rnxpm 4695 | . . . . . 6 ⊢ (∃𝑎 𝑎 ∈ A → ran (A × B) = B) | |
8 | 7 | adantl 262 | . . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ A) → ran (A × B) = B) |
9 | rnexg 4540 | . . . . . 6 ⊢ ((A × B) ∈ 𝐶 → ran (A × B) ∈ V) | |
10 | 9 | adantr 261 | . . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ A) → ran (A × B) ∈ V) |
11 | 8, 10 | eqeltrrd 2112 | . . . 4 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ A) → B ∈ V) |
12 | 6, 11 | anim12dan 532 | . . 3 ⊢ (((A × B) ∈ 𝐶 ∧ (∃𝑏 𝑏 ∈ B ∧ ∃𝑎 𝑎 ∈ A)) → (A ∈ V ∧ B ∈ V)) |
13 | 12 | ancom2s 500 | . 2 ⊢ (((A × B) ∈ 𝐶 ∧ (∃𝑎 𝑎 ∈ A ∧ ∃𝑏 𝑏 ∈ B)) → (A ∈ V ∧ B ∈ V)) |
14 | 1, 13 | sylan2br 272 | 1 ⊢ (((A × B) ∈ 𝐶 ∧ ∃x x ∈ (A × B)) → (A ∈ V ∧ B ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 × cxp 4286 dom cdm 4288 ran crn 4289 |
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-13 1401 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 ax-un 4136 |
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-rex 2306 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-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 |
This theorem is referenced by: (None) |
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