Theorem djuinr 7129

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 7159 and djufun 7170) 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 7151). (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 7122	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		dff1o5 5513	. . . . 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 7123	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		dff1o5 5513	. . . . 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 3362	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = ( ( { (/) } X. A ) i^i ( { 1o } X. B ) )
10		1n0 6490	. . . . 5 |- 1o =/= (/)
11	10	necomi 2452	. . . 4 |- (/) =/= 1o
12		disjsn2 3685	. . . 4 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
13	11, 12	ax-mp 5	. . 3 |- ( { (/) } i^i { 1o } ) = (/)
14		xpdisj1 5094	. . 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 2217	1 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)

Syntax hints:   = wceq 1364   =/= wne 2367   i^i cin 3156   (/)c0 3450   {csn 3622   X. cxp 4661   ran crn 4664   |` cres 4665   -1-1->wf1 5255   -1-1-onto->wf1o 5257   1oc1o 6467  inlcinl 7111  inrcinr 7112

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-inl 7113  df-inr 7114

This theorem is referenced by:  djuin  7130  casefun  7151  djudom  7159  djufun  7170