Theorem djuunr 7168

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 7158	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		f1ofo 5529	. . . 4 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) -> (inl |` A ) : A -onto-> ( { (/) } X. A ) )
3		forn 5501	. . . 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 7159	. . . 4 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
6		f1ofo 5529	. . . 4 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) -> (inr |` B ) : B -onto-> ( { 1o } X. B ) )
7		forn 5501	. . . 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 3325	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
10		df-dju 7140	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
11	9, 10	eqtr4i 2229	1 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )

Syntax hints:   = wceq 1373   u. cun 3164   (/)c0 3460   {csn 3633   X. cxp 4673   ran crn 4676   |` cres 4677   -onto->wfo 5269   -1-1-onto->wf1o 5270   1oc1o 6495   ⊔ cdju 7139  inlcinl 7147  inrcinr 7148

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-1o 6502  df-dju 7140  df-inl 7149  df-inr 7150

This theorem is referenced by:  djuun  7169  eldju  7170  casedm  7188  djudm  7207