Theorem updjudhf 6972

Description: The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.)

Hypotheses

Ref	Expression
updjud.f	|- ( ph -> F : A --> C )
updjud.g	|- ( ph -> G : B --> C )
updjudhf.h	|- H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) )

Assertion

Ref	Expression
updjudhf	|- ( ph -> H : ( A ⊔ B ) --> C )

Distinct variable groups:   x, A   x, B   x, C   ph, x

Allowed substitution hints:   F( x)   G( x)   H( x)

Proof of Theorem updjudhf

Step	Hyp	Ref	Expression
1		eldju2ndl 6965	. . . . . 6 |- ( ( x e. ( A ⊔ B ) /\ ( 1st ` x ) = (/) ) -> ( 2nd ` x ) e. A )
2	1	ex 114	. . . . 5 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) -> ( 2nd ` x ) e. A ) )
3		updjud.f	. . . . . 6 |- ( ph -> F : A --> C )
4		ffvelrn 5561	. . . . . . 7 |- ( ( F : A --> C /\ ( 2nd ` x ) e. A ) -> ( F ` ( 2nd ` x ) ) e. C )
5	4	ex 114	. . . . . 6 |- ( F : A --> C -> ( ( 2nd ` x ) e. A -> ( F ` ( 2nd ` x ) ) e. C ) )
6	3, 5	syl 14	. . . . 5 |- ( ph -> ( ( 2nd ` x ) e. A -> ( F ` ( 2nd ` x ) ) e. C ) )
7	2, 6	sylan9r 408	. . . 4 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( ( 1st ` x ) = (/) -> ( F ` ( 2nd ` x ) ) e. C ) )
8	7	imp 123	. . 3 |- ( ( ( ph /\ x e. ( A ⊔ B ) ) /\ ( 1st ` x ) = (/) ) -> ( F ` ( 2nd ` x ) ) e. C )
9		df-ne 2310	. . . . 5 |- ( ( 1st ` x ) =/= (/) <-> -. ( 1st ` x ) = (/) )
10		eldju2ndr 6966	. . . . . . 7 |- ( ( x e. ( A ⊔ B ) /\ ( 1st ` x ) =/= (/) ) -> ( 2nd ` x ) e. B )
11	10	ex 114	. . . . . 6 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) =/= (/) -> ( 2nd ` x ) e. B ) )
12		updjud.g	. . . . . . 7 |- ( ph -> G : B --> C )
13		ffvelrn 5561	. . . . . . . 8 |- ( ( G : B --> C /\ ( 2nd ` x ) e. B ) -> ( G ` ( 2nd ` x ) ) e. C )
14	13	ex 114	. . . . . . 7 |- ( G : B --> C -> ( ( 2nd ` x ) e. B -> ( G ` ( 2nd ` x ) ) e. C ) )
15	12, 14	syl 14	. . . . . 6 |- ( ph -> ( ( 2nd ` x ) e. B -> ( G ` ( 2nd ` x ) ) e. C ) )
16	11, 15	sylan9r 408	. . . . 5 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( ( 1st ` x ) =/= (/) -> ( G ` ( 2nd ` x ) ) e. C ) )
17	9, 16	syl5bir 152	. . . 4 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( -. ( 1st ` x ) = (/) -> ( G ` ( 2nd ` x ) ) e. C ) )
18	17	imp 123	. . 3 |- ( ( ( ph /\ x e. ( A ⊔ B ) ) /\ -. ( 1st ` x ) = (/) ) -> ( G ` ( 2nd ` x ) ) e. C )
19		eldju1st 6964	. . . . . 6 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) = 1o ) )
20		1n0 6337	. . . . . . . 8 |- 1o =/= (/)
21		neeq1 2322	. . . . . . . 8 |- ( ( 1st ` x ) = 1o -> ( ( 1st ` x ) =/= (/) <-> 1o =/= (/) ) )
22	20, 21	mpbiri 167	. . . . . . 7 |- ( ( 1st ` x ) = 1o -> ( 1st ` x ) =/= (/) )
23	22	orim2i 751	. . . . . 6 |- ( ( ( 1st ` x ) = (/) \/ ( 1st ` x ) = 1o ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) =/= (/) ) )
24	19, 23	syl 14	. . . . 5 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) =/= (/) ) )
25	24	adantl 275	. . . 4 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) =/= (/) ) )
26		dcne 2320	. . . 4 |- (DECID ( 1st ` x ) = (/) <-> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) =/= (/) ) )
27	25, 26	sylibr 133	. . 3 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> DECID ( 1st ` x ) = (/) )
28	8, 18, 27	ifcldadc 3506	. 2 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) e. C )
29		updjudhf.h	. 2 |- H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) )
30	28, 29	fmptd 5582	1 |- ( ph -> H : ( A ⊔ B ) --> C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 820   = wceq 1332   e. wcel 1481   =/= wne 2309   (/)c0 3368   ifcif 3479   |-> cmpt 3997   -->wf 5127   ` cfv 5131   1stc1st 6044   2ndc2nd 6045   1oc1o 6314   ⊔ cdju 6930

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941

This theorem is referenced by:  updjudhcoinlf  6973  updjudhcoinrg  6974  updjud  6975