Theorem updjudhf 7202

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 7195	. . . . . 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 5731	. . . . . . 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 2378	. . . . 5 |- ( ( 1st ` x ) =/= (/) <-> -. ( 1st ` x ) = (/) )
10		eldju2ndr 7196	. . . . . . 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 5731	. . . . . . . 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 7194	. . . . . 6 |- ( x e. ( A ⊔ B ) -> ( ( 1st ` x ) = (/) \/ ( 1st ` x ) = 1o ) )
20		1n0 6536	. . . . . . . 8 |- 1o =/= (/)
21		neeq1 2390	. . . . . . . 8 |- ( ( 1st ` x ) = 1o -> ( ( 1st ` x ) =/= (/) <-> 1o =/= (/) ) )
22	20, 21	mpbiri 168	. . . . . . 7 |- ( ( 1st ` x ) = 1o -> ( 1st ` x ) =/= (/) )
23	22	orim2i 763	. . . . . 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 2388	. . . 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 3605	. 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 5752	1 |- ( ph -> H : ( A ⊔ B ) --> C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2177   =/= wne 2377   (/)c0 3464   ifcif 3575   |-> cmpt 4116   -->wf 5281   ` cfv 5285   1stc1st 6242   2ndc2nd 6243   1oc1o 6513   ⊔ cdju 7160

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-1st 6244  df-2nd 6245  df-1o 6520  df-dju 7161  df-inl 7170  df-inr 7171

This theorem is referenced by:  updjudhcoinlf  7203  updjudhcoinrg  7204  updjud  7205