Theorem djuinr 7064

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 7094 and djufun 7105) 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 7086). (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 7057	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		dff1o5 5472	. . . . 5 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) <-> ( (inl |` A ) : A -1-1-> ( { (/) } X. A ) /\ ran (inl |` A ) = ( { (/) } X. A ) ) )
3	2	simprbi 275	. . . 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 7058	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		dff1o5 5472	. . . . 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 275	. . . 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 3336	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = ( ( { (/) } X. A ) i^i ( { 1o } X. B ) )
10		1n0 6435	. . . . 5 |- 1o =/= (/)
11	10	necomi 2432	. . . 4 |- (/) =/= 1o
12		disjsn2 3657	. . . 4 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
13	11, 12	ax-mp 5	. . 3 |- ( { (/) } i^i { 1o } ) = (/)
14		xpdisj1 5055	. . 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 2198	1 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)

Syntax hints:   = wceq 1353   =/= wne 2347   i^i cin 3130   (/)c0 3424   {csn 3594   X. cxp 4626   ran crn 4629   |` cres 4630   -1-1->wf1 5215   -1-1-onto->wf1o 5217   1oc1o 6412  inlcinl 7046  inrcinr 7047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-inl 7048  df-inr 7049

This theorem is referenced by:  djuin  7065  casefun  7086  djudom  7094  djufun  7105