Theorem updjudhf 7180

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 7173	. . . . . 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 5712	. . . . . . 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 2376	. . . . 5 |- ( ( 1st ` x ) =/= (/) <-> -. ( 1st ` x ) = (/) )
10		eldju2ndr 7174	. . . . . . 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 5712	. . . . . . . 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 7172	. . . . . 6 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) = 1o ) )
20		1n0 6517	. . . . . . . 8 |- 1o =/= (/)
21		neeq1 2388	. . . . . . . 8 |- ( ( 1st ` x ) = 1o -> ( ( 1st ` x ) =/= (/) <-> 1o =/= (/) ) )
22	20, 21	mpbiri 168	. . . . . . 7 |- ( ( 1st ` x ) = 1o -> ( 1st ` x ) =/= (/) )
23	22	orim2i 762	. . . . . 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 2386	. . . 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 3599	. 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 5733	1 |- ( ph -> H : ( A ⊔ B ) --> C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1372   e. wcel 2175   =/= wne 2375   (/)c0 3459   ifcif 3570   |-> cmpt 4104   -->wf 5266   ` cfv 5270   1stc1st 6223   2ndc2nd 6224   1oc1o 6494   ⊔ cdju 7138

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-1o 6501  df-dju 7139  df-inl 7148  df-inr 7149

This theorem is referenced by:  updjudhcoinlf  7181  updjudhcoinrg  7182  updjud  7183