Theorem djuinr 7261

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 7291 and djufun 7302) 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 7283). (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 7254	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		dff1o5 5592	. . . . 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 7255	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		dff1o5 5592	. . . . 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 3406	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = ( ( { (/) } X. A ) i^i ( { 1o } X. B ) )
10		1n0 6599	. . . . 5 |- 1o =/= (/)
11	10	necomi 2487	. . . 4 |- (/) =/= 1o
12		disjsn2 3732	. . . 4 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
13	11, 12	ax-mp 5	. . 3 |- ( { (/) } i^i { 1o } ) = (/)
14		xpdisj1 5161	. . 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 2252	1 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)

Syntax hints:   = wceq 1397   =/= wne 2402   i^i cin 3199   (/)c0 3494   {csn 3669   X. cxp 4723   ran crn 4726   |` cres 4727   -1-1->wf1 5323   -1-1-onto->wf1o 5325   1oc1o 6574  inlcinl 7243  inrcinr 7244

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-inl 7245  df-inr 7246

This theorem is referenced by:  djuin  7262  casefun  7283  djudom  7291  djufun  7302