Theorem updjudhf 7277

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 7270	. . . . . 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 5780	. . . . . . 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 2403	. . . . 5 |- ( ( 1st ` x ) =/= (/) <-> -. ( 1st ` x ) = (/) )
10		eldju2ndr 7271	. . . . . . 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 5780	. . . . . . . 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 7269	. . . . . 6 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) = 1o ) )
20		1n0 6599	. . . . . . . 8 |- 1o =/= (/)
21		neeq1 2415	. . . . . . . 8 |- ( ( 1st ` x ) = 1o -> ( ( 1st ` x ) =/= (/) <-> 1o =/= (/) ) )
22	20, 21	mpbiri 168	. . . . . . 7 |- ( ( 1st ` x ) = 1o -> ( 1st ` x ) =/= (/) )
23	22	orim2i 768	. . . . . 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 2413	. . . 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 3635	. 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 5801	1 |- ( ph -> H : ( A ⊔ B ) --> C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   =/= wne 2402   (/)c0 3494   ifcif 3605   |-> cmpt 4150   -->wf 5322   ` cfv 5326   1stc1st 6300   2ndc2nd 6301   1oc1o 6574   ⊔ cdju 7235

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  updjudhcoinlf  7278  updjudhcoinrg  7279  updjud  7280