Theorem djufun 7170

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 7127	. . . . 5 |- (inl |` dom F ) : dom F -1-1-> ( dom F ⊔ dom G )
3		df-f1 5263	. . . . . 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 5298	. . . 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 7128	. . . . 5 |- (inr |` dom G ) : dom G -1-1-> ( dom F ⊔ dom G )
10		df-f1 5263	. . . . . 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 5298	. . . 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 4935	. . . . . . 7 |- dom ( F o. `' (inl |` dom F ) ) C_ dom `' (inl |` dom F )
16		df-rn 4674	. . . . . . 7 |- ran (inl |` dom F ) = dom `' (inl |` dom F )
17	15, 16	sseqtrri 3218	. . . . . 6 |- dom ( F o. `' (inl |` dom F ) ) C_ ran (inl |` dom F )
18		dmcoss 4935	. . . . . . 7 |- dom ( G o. `' (inr |` dom G ) ) C_ dom `' (inr |` dom G )
19		df-rn 4674	. . . . . . 7 |- ran (inr |` dom G ) = dom `' (inr |` dom G )
20	18, 19	sseqtrri 3218	. . . . . 6 |- dom ( G o. `' (inr |` dom G ) ) C_ ran (inr |` dom G )
21		ss2in 3391	. . . . . 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 7129	. . . . . 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 3233	. . . 4 |- ( ph -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ (/) )
26		ss0 3491	. . . 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 5302	. . 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 1248	. 2 |- ( ph -> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
30		df-djud 7169	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
31	30	funeqi 5279	. 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 1364   u. cun 3155   i^i cin 3156   C_ wss 3157   (/)c0 3450   `'ccnv 4662   dom cdm 4663   ran crn 4664   |` cres 4665   o. ccom 4667   Fun wfun 5252   -->wf 5254   -1-1->wf1 5255   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112   ⊔d cdjud 7168

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114  df-djud 7169

This theorem is referenced by: (None)