Theorem djufun 7363

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 7320	. . . . 5 |- (inl |` dom F ) : dom F -1-1-> ( dom F ⊔ dom G )
3		df-f1 5338	. . . . . 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 5373	. . . 4 |- ( ( Fun F /\ Fun `' (inl |` dom F ) ) -> Fun ( F o. `' (inl |` dom F ) ) )
7	1, 5, 6	syl2anc 411	. . 3 |- ( ph -> Fun ( F o. `' (inl |` dom F ) ) )
8		djufun.g	. . . 4 |- ( ph -> Fun G )
9		inrresf1 7321	. . . . 5 |- (inr |` dom G ) : dom G -1-1-> ( dom F ⊔ dom G )
10		df-f1 5338	. . . . . 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 5373	. . . 4 |- ( ( Fun G /\ Fun `' (inr |` dom G ) ) -> Fun ( G o. `' (inr |` dom G ) ) )
14	8, 12, 13	syl2anc 411	. . 3 |- ( ph -> Fun ( G o. `' (inr |` dom G ) ) )
15		dmcoss 5008	. . . . . . 7 |- dom ( F o. `' (inl |` dom F ) ) C_ dom `' (inl |` dom F )
16		df-rn 4742	. . . . . . 7 |- ran (inl |` dom F ) = dom `' (inl |` dom F )
17	15, 16	sseqtrri 3263	. . . . . 6 |- dom ( F o. `' (inl |` dom F ) ) C_ ran (inl |` dom F )
18		dmcoss 5008	. . . . . . 7 |- dom ( G o. `' (inr |` dom G ) ) C_ dom `' (inr |` dom G )
19		df-rn 4742	. . . . . . 7 |- ran (inr |` dom G ) = dom `' (inr |` dom G )
20	18, 19	sseqtrri 3263	. . . . . 6 |- dom ( G o. `' (inr |` dom G ) ) C_ ran (inr |` dom G )
21		ss2in 3437	. . . . . 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 426	. . . . 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 7322	. . . . . 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 3278	. . . 4 |- ( ph -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ (/) )
26		ss0 3537	. . . 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 5378	. . 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 1273	. 2 |- ( ph -> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
30		df-djud 7362	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
31	30	funeqi 5354	. 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 ) )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3199   i^i cin 3200   C_ wss 3201   (/)c0 3496   `'ccnv 4730   dom cdm 4731   ran crn 4732   |` cres 4733   o. ccom 4735   Fun wfun 5327   -->wf 5329   -1-1->wf1 5330   ⊔ cdju 7296  inlcinl 7304  inrcinr 7305   ⊔d cdjud 7361

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-dju 7297  df-inl 7306  df-inr 7307  df-djud 7362

This theorem is referenced by: (None)