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Mirrors > Home > ILE Home > Th. List > dminxp | GIF version |
Description: Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) |
Ref | Expression |
---|---|
dminxp | ⊢ (dom (𝐶 ∩ (A × B)) = A ↔ ∀x ∈ A ∃y ∈ B x𝐶y) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 4470 | . . . 4 ⊢ dom (𝐶 ∩ (A × B)) = ran ◡(𝐶 ∩ (A × B)) | |
2 | cnvin 4674 | . . . . . 6 ⊢ ◡(𝐶 ∩ (A × B)) = (◡𝐶 ∩ ◡(A × B)) | |
3 | cnvxp 4685 | . . . . . . 7 ⊢ ◡(A × B) = (B × A) | |
4 | 3 | ineq2i 3129 | . . . . . 6 ⊢ (◡𝐶 ∩ ◡(A × B)) = (◡𝐶 ∩ (B × A)) |
5 | 2, 4 | eqtri 2057 | . . . . 5 ⊢ ◡(𝐶 ∩ (A × B)) = (◡𝐶 ∩ (B × A)) |
6 | 5 | rneqi 4505 | . . . 4 ⊢ ran ◡(𝐶 ∩ (A × B)) = ran (◡𝐶 ∩ (B × A)) |
7 | 1, 6 | eqtri 2057 | . . 3 ⊢ dom (𝐶 ∩ (A × B)) = ran (◡𝐶 ∩ (B × A)) |
8 | 7 | eqeq1i 2044 | . 2 ⊢ (dom (𝐶 ∩ (A × B)) = A ↔ ran (◡𝐶 ∩ (B × A)) = A) |
9 | rninxp 4707 | . 2 ⊢ (ran (◡𝐶 ∩ (B × A)) = A ↔ ∀x ∈ A ∃y ∈ B y◡𝐶x) | |
10 | vex 2554 | . . . . 5 ⊢ y ∈ V | |
11 | vex 2554 | . . . . 5 ⊢ x ∈ V | |
12 | 10, 11 | brcnv 4461 | . . . 4 ⊢ (y◡𝐶x ↔ x𝐶y) |
13 | 12 | rexbii 2325 | . . 3 ⊢ (∃y ∈ B y◡𝐶x ↔ ∃y ∈ B x𝐶y) |
14 | 13 | ralbii 2324 | . 2 ⊢ (∀x ∈ A ∃y ∈ B y◡𝐶x ↔ ∀x ∈ A ∃y ∈ B x𝐶y) |
15 | 8, 9, 14 | 3bitri 195 | 1 ⊢ (dom (𝐶 ∩ (A × B)) = A ↔ ∀x ∈ A ∃y ∈ B x𝐶y) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1242 ∀wral 2300 ∃wrex 2301 ∩ cin 2910 class class class wbr 3755 × cxp 4286 ◡ccnv 4287 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-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-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-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 |
This theorem is referenced by: (None) |
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