Theorem casef 7065

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 5350	. . . . 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 5350	. . . . 5 |- ( G : B --> X -> Fun G )
6	4, 5	syl 14	. . . 4 |- ( ph -> Fun G )
7	3, 6	casefun 7062	. . 3 |- ( ph -> Fun case ( F , G ) )
8		caserel 7064	. . . 4 |- case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) )
9		ssid 3167	. . . . 5 |- ( dom F ⊔ dom G ) C_ ( dom F ⊔ dom G )
10		frn 5356	. . . . . . 7 |- ( F : A --> X -> ran F C_ X )
11	1, 10	syl 14	. . . . . 6 |- ( ph -> ran F C_ X )
12		frn 5356	. . . . . . 7 |- ( G : B --> X -> ran G C_ X )
13	4, 12	syl 14	. . . . . 6 |- ( ph -> ran G C_ X )
14	11, 13	unssd 3303	. . . . 5 |- ( ph -> ( ran F u. ran G ) C_ X )
15		xpss12 4718	. . . . 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 412	. . . 4 |- ( ph -> ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) ) C_ ( ( dom F ⊔ dom G ) X. X ) )
17	8, 16	sstrid 3158	. . 3 |- ( ph -> case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. X ) )
18		funssxp 5367	. . . 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 7063	. . . 4 |- dom case ( F , G ) = ( dom F ⊔ dom G )
22		fdm 5353	. . . . . 6 |- ( F : A --> X -> dom F = A )
23	1, 22	syl 14	. . . . 5 |- ( ph -> dom F = A )
24		fdm 5353	. . . . . 6 |- ( G : B --> X -> dom G = B )
25	4, 24	syl 14	. . . . 5 |- ( ph -> dom G = B )
26		djueq12 7016	. . . . 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	eqtrid 2215	. . 3 |- ( ph -> dom case ( F , G ) = ( A ⊔ B ) )
29	28	feq2d 5335	. 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 1348   u. cun 3119   C_ wss 3121   X. cxp 4609   dom cdm 4611   ran crn 4612   Fun wfun 5192   -->wf 5194   ⊔ cdju 7014  casecdjucase 7060

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025  df-case 7061

This theorem is referenced by:  casef1  7067  omp1eomlem  7071  ctm  7086