Theorem djuinr 6956

Description: The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 6986 and djufun 6997) while the simpler statement |- ( ran inl i^i ran inr ) = (/) is easily recovered from it by substituting _V for both A and B as done in casefun 6978). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)

Assertion

Ref	Expression
djuinr	|- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)

Proof of Theorem djuinr

Step	Hyp	Ref	Expression
1		djulf1or 6949	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		dff1o5 5384	. . . . 5 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) <-> ( (inl |` A ) : A -1-1-> ( { (/) } X. A ) /\ ran (inl |` A ) = ( { (/) } X. A ) ) )
3	2	simprbi 273	. . . 4 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) -> ran (inl |` A ) = ( { (/) } X. A ) )
4	1, 3	ax-mp 5	. . 3 |- ran (inl |` A ) = ( { (/) } X. A )
5		djurf1or 6950	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		dff1o5 5384	. . . . 5 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) <-> ( (inr |` B ) : B -1-1-> ( { 1o } X. B ) /\ ran (inr |` B ) = ( { 1o } X. B ) ) )
7	6	simprbi 273	. . . 4 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) -> ran (inr |` B ) = ( { 1o } X. B ) )
8	5, 7	ax-mp 5	. . 3 |- ran (inr |` B ) = ( { 1o } X. B )
9	4, 8	ineq12i 3280	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = ( ( { (/) } X. A ) i^i ( { 1o } X. B ) )
10		1n0 6337	. . . . 5 |- 1o =/= (/)
11	10	necomi 2394	. . . 4 |- (/) =/= 1o
12		disjsn2 3594	. . . 4 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
13	11, 12	ax-mp 5	. . 3 |- ( { (/) } i^i { 1o } ) = (/)
14		xpdisj1 4971	. . 3 |- ( ( { (/) } i^i { 1o } ) = (/) -> ( ( { (/) } X. A ) i^i ( { 1o } X. B ) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( ( { (/) } X. A ) i^i ( { 1o } X. B ) ) = (/)
16	9, 15	eqtri 2161	1 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)

Syntax hints:   = wceq 1332   =/= wne 2309   i^i cin 3075   (/)c0 3368   {csn 3532   X. cxp 4545   ran crn 4548   |` cres 4549   -1-1->wf1 5128   -1-1-onto->wf1o 5130   1oc1o 6314  inlcinl 6938  inrcinr 6939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-inl 6940  df-inr 6941

This theorem is referenced by:  djuin  6957  casefun  6978  djudom  6986  djufun  6997