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Mirrors > Home > ILE Home > Th. List > rnxpm | GIF version |
Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.) |
Ref | Expression |
---|---|
rnxpm | ⊢ (∃x x ∈ A → ran (A × B) = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 4299 | . . 3 ⊢ ran (A × B) = dom ◡(A × B) | |
2 | cnvxp 4685 | . . . 4 ⊢ ◡(A × B) = (B × A) | |
3 | 2 | dmeqi 4479 | . . 3 ⊢ dom ◡(A × B) = dom (B × A) |
4 | 1, 3 | eqtri 2057 | . 2 ⊢ ran (A × B) = dom (B × A) |
5 | dmxpm 4498 | . 2 ⊢ (∃x x ∈ A → dom (B × A) = B) | |
6 | 4, 5 | syl5eq 2081 | 1 ⊢ (∃x x ∈ A → ran (A × B) = B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∃wex 1378 ∈ wcel 1390 × 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 |
This theorem is referenced by: ssxpbm 4699 ssxp2 4701 xpexr2m 4705 xpima2m 4711 unixpm 4796 |
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