Theorem casef 7428

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 5536	. . . . 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 5536	. . . . 5 |- ( G : B --> X -> Fun G )
6	4, 5	syl 14	. . . 4 |- ( ph -> Fun G )
7	3, 6	casefun 7425	. . 3 |- ( ph -> Fun case ( F , G ) )
8		caserel 7427	. . . 4 |- case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) )
9		ssid 3268	. . . . 5 |- ( dom F ⊔ dom G ) C_ ( dom F ⊔ dom G )
10		frn 5542	. . . . . . 7 |- ( F : A --> X -> ran F C_ X )
11	1, 10	syl 14	. . . . . 6 |- ( ph -> ran F C_ X )
12		frn 5542	. . . . . . 7 |- ( G : B --> X -> ran G C_ X )
13	4, 12	syl 14	. . . . . 6 |- ( ph -> ran G C_ X )
14	11, 13	unssd 3405	. . . . 5 |- ( ph -> ( ran F u. ran G ) C_ X )
15		xpss12 4882	. . . . 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 418	. . . 4 |- ( ph -> ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) ) C_ ( ( dom F ⊔ dom G ) X. X ) )
17	8, 16	sstrid 3259	. . 3 |- ( ph -> case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. X ) )
18		funssxp 5557	. . . 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 415	. 2 |- ( ph -> case ( F , G ) : dom case ( F , G ) --> X )
21		casedm 7426	. . . 4 |- dom case ( F , G ) = ( dom F ⊔ dom G )
22		fdm 5539	. . . . . 6 |- ( F : A --> X -> dom F = A )
23	1, 22	syl 14	. . . . 5 |- ( ph -> dom F = A )
24		fdm 5539	. . . . . 6 |- ( G : B --> X -> dom G = B )
25	4, 24	syl 14	. . . . 5 |- ( ph -> dom G = B )
26		djueq12 7379	. . . . 5 |- ( ( dom F = A /\ dom G = B ) -> ( dom F ⊔ dom G ) = ( A ⊔ B ) )
27	23, 25, 26	syl2anc 415	. . . 4 |- ( ph -> ( dom F ⊔ dom G ) = ( A ⊔ B ) )
28	21, 27	eqtrid 2283	. . 3 |- ( ph -> dom case ( F , G ) = ( A ⊔ B ) )
29	28	feq2d 5521	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   u. cun 3218   C_ wss 3220   X. cxp 4772   dom cdm 4774   ran crn 4775   Fun wfun 5371   -->wf 5373   ⊔ cdju 7377  casecdjucase 7423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375  df-1o 6687  df-dju 7378  df-inl 7387  df-inr 7388  df-case 7424

This theorem is used by:  casef1  7430  omp1eomlem  7434  ctm  7449