Theorem djuunr 7060

Description: The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)

Assertion

Ref	Expression
djuunr	|- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )

Proof of Theorem djuunr

Step	Hyp	Ref	Expression
1		djulf1or 7050	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		f1ofo 5465	. . . 4 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) -> (inl |` A ) : A -onto-> ( { (/) } X. A ) )
3		forn 5438	. . . 4 |- ( (inl |` A ) : A -onto-> ( { (/) } X. A ) -> ran (inl |` A ) = ( { (/) } X. A ) )
4	1, 2, 3	mp2b 8	. . 3 |- ran (inl |` A ) = ( { (/) } X. A )
5		djurf1or 7051	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		f1ofo 5465	. . . 4 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) -> (inr |` B ) : B -onto-> ( { 1o } X. B ) )
7		forn 5438	. . . 4 |- ( (inr |` B ) : B -onto-> ( { 1o } X. B ) -> ran (inr |` B ) = ( { 1o } X. B ) )
8	5, 6, 7	mp2b 8	. . 3 |- ran (inr |` B ) = ( { 1o } X. B )
9	4, 8	uneq12i 3287	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
10		df-dju 7032	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
11	9, 10	eqtr4i 2201	1 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )

Syntax hints:   = wceq 1353   u. cun 3127   (/)c0 3422   {csn 3592   X. cxp 4622   ran crn 4625   |` cres 4626   -onto->wfo 5211   -1-1-onto->wf1o 5212   1oc1o 6405   ⊔ cdju 7031  inlcinl 7039  inrcinr 7040

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 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-iord 4364  df-on 4366  df-suc 4369  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-1st 6136  df-2nd 6137  df-1o 6412  df-dju 7032  df-inl 7041  df-inr 7042

This theorem is referenced by:  djuun  7061  eldju  7062  casedm  7080  djudm  7099