Theorem djuunr 7264

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 7254	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		f1ofo 5590	. . . 4 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) -> (inl |` A ) : A -onto-> ( { (/) } X. A ) )
3		forn 5562	. . . 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 7255	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		f1ofo 5590	. . . 4 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) -> (inr |` B ) : B -onto-> ( { 1o } X. B ) )
7		forn 5562	. . . 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 3359	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
10		df-dju 7236	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
11	9, 10	eqtr4i 2255	1 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )

Syntax hints:   = wceq 1397   u. cun 3198   (/)c0 3494   {csn 3669   X. cxp 4723   ran crn 4726   |` cres 4727   -onto->wfo 5324   -1-1-onto->wf1o 5325   1oc1o 6574   ⊔ cdju 7235  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-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  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-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  djuun  7265  eldju  7266  casedm  7284  djudm  7303