Theorem djufun 7444

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 7401	. . . . 5 |- (inl |` dom F ) : dom F -1-1-> ( dom F ⊔ dom G )
3		df-f1 5382	. . . . . 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 275	. . . . 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 5417	. . . 4 |- ( ( Fun F /\ Fun `' (inl |` dom F ) ) -> Fun ( F o. `' (inl |` dom F ) ) )
7	1, 5, 6	syl2anc 415	. . 3 |- ( ph -> Fun ( F o. `' (inl |` dom F ) ) )
8		djufun.g	. . . 4 |- ( ph -> Fun G )
9		inrresf1 7402	. . . . 5 |- (inr |` dom G ) : dom G -1-1-> ( dom F ⊔ dom G )
10		df-f1 5382	. . . . . 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 275	. . . . 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 5417	. . . 4 |- ( ( Fun G /\ Fun `' (inr |` dom G ) ) -> Fun ( G o. `' (inr |` dom G ) ) )
14	8, 12, 13	syl2anc 415	. . 3 |- ( ph -> Fun ( G o. `' (inr |` dom G ) ) )
15		dmcoss 5052	. . . . . . 7 |- dom ( F o. `' (inl |` dom F ) ) C_ dom `' (inl |` dom F )
16		df-rn 4785	. . . . . . 7 |- ran (inl |` dom F ) = dom `' (inl |` dom F )
17	15, 16	sseqtrri 3283	. . . . . 6 |- dom ( F o. `' (inl |` dom F ) ) C_ ran (inl |` dom F )
18		dmcoss 5052	. . . . . . 7 |- dom ( G o. `' (inr |` dom G ) ) C_ dom `' (inr |` dom G )
19		df-rn 4785	. . . . . . 7 |- ran (inr |` dom G ) = dom `' (inr |` dom G )
20	18, 19	sseqtrri 3283	. . . . . 6 |- dom ( G o. `' (inr |` dom G ) ) C_ ran (inr |` dom G )
21		ss2in 3459	. . . . . 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 430	. . . . 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 7403	. . . . . 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 3298	. . . 4 |- ( ph -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ (/) )
26		ss0 3563	. . . 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 5422	. . 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 1277	. 2 |- ( ph -> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
30		df-djud 7443	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
31	30	funeqi 5398	. 2 |- ( Fun ( F ⊔d G ) <-> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
32	29, 31	sylibr 134	1 |- ( ph -> Fun ( F ⊔d G ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   `'ccnv 4773   dom cdm 4774   ran crn 4775   |` cres 4776   o. ccom 4778   Fun wfun 5371   -->wf 5373   -1-1->wf1 5374   ⊔ cdju 7377  inlcinl 7385  inrcinr 7386   ⊔d cdjud 7442

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375  df-1o 6687  df-dju 7378  df-inl 7387  df-inr 7388  df-djud 7443

This theorem is used by: (None)