Theorem casef 7255

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 5476	. . . . 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 5476	. . . . 5 |- ( G : B --> X -> Fun G )
6	4, 5	syl 14	. . . 4 |- ( ph -> Fun G )
7	3, 6	casefun 7252	. . 3 |- ( ph -> Fun case ( F , G ) )
8		caserel 7254	. . . 4 |- case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) )
9		ssid 3244	. . . . 5 |- ( dom F ⊔ dom G ) C_ ( dom F ⊔ dom G )
10		frn 5482	. . . . . . 7 |- ( F : A --> X -> ran F C_ X )
11	1, 10	syl 14	. . . . . 6 |- ( ph -> ran F C_ X )
12		frn 5482	. . . . . . 7 |- ( G : B --> X -> ran G C_ X )
13	4, 12	syl 14	. . . . . 6 |- ( ph -> ran G C_ X )
14	11, 13	unssd 3380	. . . . 5 |- ( ph -> ( ran F u. ran G ) C_ X )
15		xpss12 4826	. . . . 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 414	. . . 4 |- ( ph -> ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) ) C_ ( ( dom F ⊔ dom G ) X. X ) )
17	8, 16	sstrid 3235	. . 3 |- ( ph -> case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. X ) )
18		funssxp 5493	. . . 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 274	. . 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 411	. 2 |- ( ph -> case ( F , G ) : dom case ( F , G ) --> X )
21		casedm 7253	. . . 4 |- dom case ( F , G ) = ( dom F ⊔ dom G )
22		fdm 5479	. . . . . 6 |- ( F : A --> X -> dom F = A )
23	1, 22	syl 14	. . . . 5 |- ( ph -> dom F = A )
24		fdm 5479	. . . . . 6 |- ( G : B --> X -> dom G = B )
25	4, 24	syl 14	. . . . 5 |- ( ph -> dom G = B )
26		djueq12 7206	. . . . 5 |- ( ( dom F = A /\ dom G = B ) -> ( dom F ⊔ dom G ) = ( A ⊔ B ) )
27	23, 25, 26	syl2anc 411	. . . 4 |- ( ph -> ( dom F ⊔ dom G ) = ( A ⊔ B ) )
28	21, 27	eqtrid 2274	. . 3 |- ( ph -> dom case ( F , G ) = ( A ⊔ B ) )
29	28	feq2d 5461	. 2 |- ( ph -> (case ( F , G ) : dom case ( F , G ) --> X <-> case ( F , G ) : ( A ⊔ B ) --> X ) )
30	20, 29	mpbid 147	1 |- ( ph -> case ( F , G ) : ( A ⊔ B ) --> X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   u. cun 3195   C_ wss 3197   X. cxp 4717   dom cdm 4719   ran crn 4720   Fun wfun 5312   -->wf 5314   ⊔ cdju 7204  casecdjucase 7250

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-dju 7205  df-inl 7214  df-inr 7215  df-case 7251

This theorem is referenced by:  casef1  7257  omp1eomlem  7261  ctm  7276