Theorem casef 7190

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 5428	. . . . 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 5428	. . . . 5 |- ( G : B --> X -> Fun G )
6	4, 5	syl 14	. . . 4 |- ( ph -> Fun G )
7	3, 6	casefun 7187	. . 3 |- ( ph -> Fun case ( F , G ) )
8		caserel 7189	. . . 4 |- case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. ( ran F u. ran G ) )
9		ssid 3213	. . . . 5 |- ( dom F ⊔ dom G ) C_ ( dom F ⊔ dom G )
10		frn 5434	. . . . . . 7 |- ( F : A --> X -> ran F C_ X )
11	1, 10	syl 14	. . . . . 6 |- ( ph -> ran F C_ X )
12		frn 5434	. . . . . . 7 |- ( G : B --> X -> ran G C_ X )
13	4, 12	syl 14	. . . . . 6 |- ( ph -> ran G C_ X )
14	11, 13	unssd 3349	. . . . 5 |- ( ph -> ( ran F u. ran G ) C_ X )
15		xpss12 4782	. . . . 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 3204	. . 3 |- ( ph -> case ( F , G ) C_ ( ( dom F ⊔ dom G ) X. X ) )
18		funssxp 5445	. . . 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 7188	. . . 4 |- dom case ( F , G ) = ( dom F ⊔ dom G )
22		fdm 5431	. . . . . 6 |- ( F : A --> X -> dom F = A )
23	1, 22	syl 14	. . . . 5 |- ( ph -> dom F = A )
24		fdm 5431	. . . . . 6 |- ( G : B --> X -> dom G = B )
25	4, 24	syl 14	. . . . 5 |- ( ph -> dom G = B )
26		djueq12 7141	. . . . 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 2250	. . 3 |- ( ph -> dom case ( F , G ) = ( A ⊔ B ) )
29	28	feq2d 5413	. 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 1373   u. cun 3164   C_ wss 3166   X. cxp 4673   dom cdm 4675   ran crn 4676   Fun wfun 5265   -->wf 5267   ⊔ cdju 7139  casecdjucase 7185

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-1o 6502  df-dju 7140  df-inl 7149  df-inr 7150  df-case 7186

This theorem is referenced by:  casef1  7192  omp1eomlem  7196  ctm  7211