Theorem djufun 7093

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 7050	. . . . 5 |- (inl |` dom F ) : dom F -1-1-> ( dom F ⊔ dom G )
3		df-f1 5213	. . . . . 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 5248	. . . 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 7051	. . . . 5 |- (inr |` dom G ) : dom G -1-1-> ( dom F ⊔ dom G )
10		df-f1 5213	. . . . . 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 5248	. . . 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 4889	. . . . . . 7 |- dom ( F o. `' (inl |` dom F ) ) C_ dom `' (inl |` dom F )
16		df-rn 4631	. . . . . . 7 |- ran (inl |` dom F ) = dom `' (inl |` dom F )
17	15, 16	sseqtrri 3188	. . . . . 6 |- dom ( F o. `' (inl |` dom F ) ) C_ ran (inl |` dom F )
18		dmcoss 4889	. . . . . . 7 |- dom ( G o. `' (inr |` dom G ) ) C_ dom `' (inr |` dom G )
19		df-rn 4631	. . . . . . 7 |- ran (inr |` dom G ) = dom `' (inr |` dom G )
20	18, 19	sseqtrri 3188	. . . . . 6 |- dom ( G o. `' (inr |` dom G ) ) C_ ran (inr |` dom G )
21		ss2in 3361	. . . . . 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 7052	. . . . . 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 3203	. . . 4 |- ( ph -> ( dom ( F o. `' (inl |` dom F ) ) i^i dom ( G o. `' (inr |` dom G ) ) ) C_ (/) )
26		ss0 3461	. . . 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 5252	. . 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 1237	. 2 |- ( ph -> Fun ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) )
30		df-djud 7092	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
31	30	funeqi 5229	. 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 1353   u. cun 3125   i^i cin 3126   C_ wss 3127   (/)c0 3420   `'ccnv 4619   dom cdm 4620   ran crn 4621   |` cres 4622   o. ccom 4624   Fun wfun 5202   -->wf 5204   -1-1->wf1 5205   ⊔ cdju 7026  inlcinl 7034  inrcinr 7035   ⊔d cdjud 7091

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037  df-djud 7092

This theorem is referenced by: (None)