Theorem updjudhf 7242

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 7235	. . . . . 6 |- ( ( x e. ( A ⊔ B ) /\ ( 1st ` x ) = (/) ) -> ( 2nd ` x ) e. A )
2	1	ex 115	. . . . 5 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) -> ( 2nd ` x ) e. A ) )
3		updjud.f	. . . . . 6 |- ( ph -> F : A --> C )
4		ffvelcdm 5767	. . . . . . 7 |- ( ( F : A --> C /\ ( 2nd ` x ) e. A ) -> ( F ` ( 2nd ` x ) ) e. C )
5	4	ex 115	. . . . . 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 410	. . . 4 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( ( 1st ` x ) = (/) -> ( F ` ( 2nd ` x ) ) e. C ) )
8	7	imp 124	. . 3 |- ( ( ( ph /\ x e. ( A ⊔ B ) ) /\ ( 1st ` x ) = (/) ) -> ( F ` ( 2nd ` x ) ) e. C )
9		df-ne 2401	. . . . 5 |- ( ( 1st ` x ) =/= (/) <-> -. ( 1st ` x ) = (/) )
10		eldju2ndr 7236	. . . . . . 7 |- ( ( x e. ( A ⊔ B ) /\ ( 1st ` x ) =/= (/) ) -> ( 2nd ` x ) e. B )
11	10	ex 115	. . . . . 6 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) =/= (/) -> ( 2nd ` x ) e. B ) )
12		updjud.g	. . . . . . 7 |- ( ph -> G : B --> C )
13		ffvelcdm 5767	. . . . . . . 8 |- ( ( G : B --> C /\ ( 2nd ` x ) e. B ) -> ( G ` ( 2nd ` x ) ) e. C )
14	13	ex 115	. . . . . . 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 410	. . . . 5 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( ( 1st ` x ) =/= (/) -> ( G ` ( 2nd ` x ) ) e. C ) )
17	9, 16	biimtrrid 153	. . . 4 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( -. ( 1st ` x ) = (/) -> ( G ` ( 2nd ` x ) ) e. C ) )
18	17	imp 124	. . 3 |- ( ( ( ph /\ x e. ( A ⊔ B ) ) /\ -. ( 1st ` x ) = (/) ) -> ( G ` ( 2nd ` x ) ) e. C )
19		eldju1st 7234	. . . . . 6 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) = 1o ) )
20		1n0 6576	. . . . . . . 8 |- 1o =/= (/)
21		neeq1 2413	. . . . . . . 8 |- ( ( 1st ` x ) = 1o -> ( ( 1st ` x ) =/= (/) <-> 1o =/= (/) ) )
22	20, 21	mpbiri 168	. . . . . . 7 |- ( ( 1st ` x ) = 1o -> ( 1st ` x ) =/= (/) )
23	22	orim2i 766	. . . . . 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 277	. . . 4 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) =/= (/) ) )
26		dcne 2411	. . . 4 |- (DECID ( 1st ` x ) = (/) <-> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) =/= (/) ) )
27	25, 26	sylibr 134	. . 3 |- ( ( ph /\ x e. ( A ⊔ B ) ) -> DECID ( 1st ` x ) = (/) )
28	8, 18, 27	ifcldadc 3632	. 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 5788	1 |- ( ph -> H : ( A ⊔ B ) --> C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   ifcif 3602   |-> cmpt 4144   -->wf 5313   ` cfv 5317   1stc1st 6282   2ndc2nd 6283   1oc1o 6553   ⊔ cdju 7200

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  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-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1st 6284  df-2nd 6285  df-1o 6560  df-dju 7201  df-inl 7210  df-inr 7211

This theorem is referenced by:  updjudhcoinlf  7243  updjudhcoinrg  7244  updjud  7245