Theorem casef 7053

Description: The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
casef.f	|- ( ph -> F : A --> X )
casef.g	|- ( ph -> G : B --> X )

Assertion

Ref	Expression
casef	|- ( ph -> case ( F , G ) : ( A ⊔ B ) --> X )

Proof of Theorem casef

Step	Hyp	Ref	Expression
1		casef.f	. . . . 5 |- ( ph -> F : A --> X )
2		ffun 5340	. . . . 5 |- ( F : A --> X -> Fun F )
3	1, 2	syl 14	. . . 4 |- ( ph -> Fun F )
4		casef.g	. . . . 5 |- ( ph -> G : B --> X )
5		ffun 5340	. . . . 5 |- ( G : B --> X -> Fun G )
6	4, 5	syl 14	. . . 4 |- ( ph -> Fun G )
7	3, 6	casefun 7050	. . 3 |- ( ph -> Fun case ( F , G ) )
8		caserel 7052	. . . 4 |- case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) )
9		ssid 3162	. . . . 5 |- ( dom F ⊔ dom G ) C_ ( dom F ⊔ dom G )
10		frn 5346	. . . . . . 7 |- ( F : A --> X -> ran F C_ X )
11	1, 10	syl 14	. . . . . 6 |- ( ph -> ran F C_ X )
12		frn 5346	. . . . . . 7 |- ( G : B --> X -> ran G C_ X )
13	4, 12	syl 14	. . . . . 6 |- ( ph -> ran G C_ X )
14	11, 13	unssd 3298	. . . . 5 |- ( ph -> ( ran F u. ran G ) C_ X )
15		xpss12 4711	. . . . 5 |- ( ( ( dom F ⊔ dom G ) C_ ( dom F ⊔ dom G ) /\ ( ran F u. ran G ) C_ X ) -> ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) ) C_ ( ( dom F ⊔ dom G ) X. X ) )
16	9, 14, 15	sylancr 411	. . . 4 |- ( ph -> ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) ) C_ ( ( dom F ⊔ dom G ) X. X ) )
17	8, 16	sstrid 3153	. . 3 |- ( ph -> case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. X ) )
18		funssxp 5357	. . . 4 |- ( ( Fun case ( F , G ) /\ case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. X ) ) <-> (case ( F , G ) : dom case ( F , G ) --> X /\ dom case ( F , G ) C_ ( dom F ⊔ dom G ) ) )
19	18	simplbi 272	. . 3 |- ( ( Fun case ( F , G ) /\ case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. X ) ) -> case ( F , G ) : dom case ( F , G ) --> X )
20	7, 17, 19	syl2anc 409	. 2 |- ( ph -> case ( F , G ) : dom case ( F , G ) --> X )
21		casedm 7051	. . . 4 |- dom case ( F , G ) = ( dom F ⊔ dom G )
22		fdm 5343	. . . . . 6 |- ( F : A --> X -> dom F = A )
23	1, 22	syl 14	. . . . 5 |- ( ph -> dom F = A )
24		fdm 5343	. . . . . 6 |- ( G : B --> X -> dom G = B )
25	4, 24	syl 14	. . . . 5 |- ( ph -> dom G = B )
26		djueq12 7004	. . . . 5 |- ( ( dom F = A /\ dom G = B ) -> ( dom F ⊔ dom G ) = ( A ⊔ B ) )
27	23, 25, 26	syl2anc 409	. . . 4 |- ( ph -> ( dom F ⊔ dom G ) = ( A ⊔ B ) )
28	21, 27	syl5eq 2211	. . 3 |- ( ph -> dom case ( F , G ) = ( A ⊔ B ) )
29	28	feq2d 5325	. 2 |- ( ph -> (case ( F , G ) : dom case ( F , G ) --> X <-> case ( F , G ) : ( A ⊔ B ) --> X ) )
30	20, 29	mpbid 146	1 |- ( ph -> case ( F , G ) : ( A ⊔ B ) --> X )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   u. cun 3114   C_ wss 3116   X. cxp 4602   dom cdm 4604   ran crn 4605   Fun wfun 5182   -->wf 5184   ⊔ cdju 7002  casecdjucase 7048

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013  df-case 7049

This theorem is referenced by:  casef1  7055  omp1eomlem  7059  ctm  7074