Theorem djufun 6989

Description: The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
djufun.f	|- ( ph -> Fun F )
djufun.g	|- ( ph -> Fun G )

Assertion

Ref	Expression
djufun	|- ( ph -> Fun ( F ⊔d G ) )

Proof of Theorem djufun

Step	Hyp	Ref	Expression
1		djufun.f	. . . 4 |- ( ph -> Fun F )
2		inlresf1 6946	. . . . 5 |- (inl |` dom F ) : dom F -1-1-> ( dom F ⊔ dom G )
3		df-f1 5128	. . . . . 6 |- ( (inl |` dom F ) : dom F -1-1-> ( dom F ⊔ dom G ) <-> ( (inl |` dom F ) : dom F --> ( dom F ⊔ dom G ) /\ Fun `' (inl |` dom F ) ) )
4	3	simprbi 273	. . . . 5 |- ( (inl |` dom F ) : dom F -1-1-> ( dom F ⊔ dom G ) -> Fun `' (inl |` dom F ) )
5	2, 4	mp1i 10	. . . 4 |- ( ph -> Fun `' (inl |` dom F ) )
6		funco 5163	. . . 4 |- ( ( Fun F /\ Fun `' (inl |` dom F ) ) -> Fun ( F o. `' (inl |` dom F ) ) )
7	1, 5, 6	syl2anc 408	. . 3 |- ( ph -> Fun ( F o. `' (inl |` dom F ) ) )
8		djufun.g	. . . 4 |- ( ph -> Fun G )
9		inrresf1 6947	. . . . 5 |- (inr |` dom G ) : dom G -1-1-> ( dom F ⊔ dom G )
10		df-f1 5128	. . . . . 6 |- ( (inr |` dom G ) : dom G -1-1-> ( dom F ⊔ dom G ) <-> ( (inr |` dom G ) : dom G --> ( dom F ⊔ dom G ) /\ Fun `' (inr |` dom G ) ) )
11	10	simprbi 273	. . . . 5 |- ( (inr |` dom G ) : dom G -1-1-> ( dom F ⊔ dom G ) -> Fun `' (inr |` dom G ) )
12	9, 11	mp1i 10	. . . 4 |- ( ph -> Fun `' (inr |` dom G ) )
13		funco 5163	. . . 4 |- ( ( Fun G /\ Fun `' (inr |` dom G ) ) -> Fun ( G o. `' (inr |` dom G ) ) )
14	8, 12, 13	syl2anc 408	. . 3 |- ( ph -> Fun ( G o. `' (inr |` dom G ) ) )
15		dmcoss 4808	. . . . . . 7 |- dom ( F o. `' (inl |` dom F ) ) C_ dom `' (inl |` dom F )
16		df-rn 4550	. . . . . . 7 |- ran (inl |` dom F ) = dom `' (inl |` dom F )
17	15, 16	sseqtrri 3132	. . . . . 6 |- dom ( F o. `' (inl |` dom F ) ) C_ ran (inl |` dom F )
18		dmcoss 4808	. . . . . . 7 |- dom ( G o. `' (inr |` dom G ) ) C_ dom `' (inr |` dom G )
19		df-rn 4550	. . . . . . 7 |- ran (inr |` dom G ) = dom `' (inr |` dom G )
20	18, 19	sseqtrri 3132	. . . . . 6 |- dom ( G o. `' (inr |` dom G ) ) C_ ran (inr |` dom G )
21		ss2in 3304	. . . . . 6 |- ( ( dom ( F o. `' (inl |` dom F ) ) C_ ran (inl |` dom F ) /\ dom ( G o. `' (inr |` dom G ) ) C_ ran (inr |` dom G ) ) -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ ( ran (inl |` dom F ) i^i ran (inr |` dom G ) ) )
22	17, 20, 21	mp2an 422	. . . . 5 |- ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ ( ran (inl |` dom F ) i^i ran (inr |` dom G ) )
23		djuinr 6948	. . . . . 6 |- ( ran (inl |` dom F ) i^i ran (inr |` dom G ) ) = (/)
24	23	a1i 9	. . . . 5 |- ( ph -> ( ran (inl |` dom F ) i^i ran (inr |` dom G ) ) = (/) )
25	22, 24	sseqtrid 3147	. . . 4 |- ( ph -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ (/) )
26		ss0 3403	. . . 4 |- ( ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ (/) -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) = (/) )
27	25, 26	syl 14	. . 3 |- ( ph -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) = (/) )
28		funun 5167	. . 3 |- ( ( ( Fun ( F o. `' (inl |` dom F ) ) /\ Fun ( G o. `' (inr |` dom G ) ) ) /\ ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) = (/) ) -> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
29	7, 14, 27, 28	syl21anc 1215	. 2 |- ( ph -> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
30		df-djud 6988	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
31	30	funeqi 5144	. 2 |- ( Fun ( F ⊔d G ) <-> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
32	29, 31	sylibr 133	1 |- ( ph -> Fun ( F ⊔d G ) )

Syntax hints:   -> wi 4   = wceq 1331   u. cun 3069   i^i cin 3070   C_ wss 3071   (/)c0 3363   `'ccnv 4538   dom cdm 4539   ran crn 4540   |` cres 4541   o. ccom 4543   Fun wfun 5117   -->wf 5119   -1-1->wf1 5120   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931   ⊔d cdjud 6987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933  df-djud 6988

This theorem is referenced by: (None)