Theorem djuinr 6948

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 6978 and djufun 6989) 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 6970). (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 6941	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		dff1o5 5376	. . . . 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 6942	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		dff1o5 5376	. . . . 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 3275	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = ( ( { (/) } X. A ) i^i ( { 1o } X. B ) )
10		1n0 6329	. . . . 5 |- 1o =/= (/)
11	10	necomi 2393	. . . 4 |- (/) =/= 1o
12		disjsn2 3586	. . . 4 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
13	11, 12	ax-mp 5	. . 3 |- ( { (/) } i^i { 1o } ) = (/)
14		xpdisj1 4963	. . 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 2160	1 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)

Syntax hints:   = wceq 1331   =/= wne 2308   i^i cin 3070   (/)c0 3363   {csn 3527   X. cxp 4537   ran crn 4540   |` cres 4541   -1-1->wf1 5120   -1-1-onto->wf1o 5122   1oc1o 6306  inlcinl 6930  inrcinr 6931

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-inl 6932  df-inr 6933

This theorem is referenced by:  djuin  6949  casefun  6970  djudom  6978  djufun  6989