![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rnxpss | GIF version |
Description: The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
rnxpss | ⊢ ran (A × B) ⊆ B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 4299 | . 2 ⊢ 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 | dmxpss 4696 | . . 3 ⊢ dom (B × A) ⊆ B | |
5 | 3, 4 | eqsstri 2969 | . 2 ⊢ dom ◡(A × B) ⊆ B |
6 | 1, 5 | eqsstri 2969 | 1 ⊢ ran (A × B) ⊆ B |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 2911 × 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: rnxpid 4698 ssxpbm 4699 ssxp2 4701 ssrnres 4706 funssxp 5003 fconst 5025 dff2 5254 fliftf 5382 |
Copyright terms: Public domain | W3C validator |