Theorem updjudhcoinlf 7410

Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-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
updjudhcoinlf	|- ( ph -> ( H o. (inl |` A ) ) = F )

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

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

Proof of Theorem updjudhcoinlf

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		updjud.f	. . . . 5 |- ( ph -> F : A --> C )
2		updjud.g	. . . . 5 |- ( ph -> G : B --> C )
3		updjudhf.h	. . . . 5 |- H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) )
4	1, 2, 3	updjudhf 7409	. . . 4 |- ( ph -> H : ( A ⊔ B ) --> C )
5		ffn 5528	. . . 4 |- ( H : ( A ⊔ B ) --> C -> H Fn ( A ⊔ B ) )
6	4, 5	syl 14	. . 3 |- ( ph -> H Fn ( A ⊔ B ) )
7		inlresf1 7391	. . . 4 |- (inl |` A ) : A -1-1-> ( A ⊔ B )
8		f1fn 5595	. . . 4 |- ( (inl |` A ) : A -1-1-> ( A ⊔ B ) -> (inl |` A ) Fn A )
9	7, 8	mp1i 10	. . 3 |- ( ph -> (inl |` A ) Fn A )
10		f1f 5593	. . . . 5 |- ( (inl |` A ) : A -1-1-> ( A ⊔ B ) -> (inl |` A ) : A --> ( A ⊔ B ) )
11	7, 10	ax-mp 5	. . . 4 |- (inl |` A ) : A --> ( A ⊔ B )
12		frn 5537	. . . 4 |- ( (inl |` A ) : A --> ( A ⊔ B ) -> ran (inl |` A ) C_ ( A ⊔ B ) )
13	11, 12	mp1i 10	. . 3 |- ( ph -> ran (inl |` A ) C_ ( A ⊔ B ) )
14		fnco 5486	. . 3 |- ( ( H Fn ( A ⊔ B ) /\ (inl |` A ) Fn A /\ ran (inl |` A ) C_ ( A ⊔ B ) ) -> ( H o. (inl |` A ) ) Fn A )
15	6, 9, 13, 14	syl3anc 1278	. 2 |- ( ph -> ( H o. (inl |` A ) ) Fn A )
16		ffn 5528	. . 3 |- ( F : A --> C -> F Fn A )
17	1, 16	syl 14	. 2 |- ( ph -> F Fn A )
18		fvco2 5768	. . . 4 |- ( ( (inl |` A ) Fn A /\ a e. A ) -> ( ( H o. (inl |` A ) ) ` a ) = ( H ` ( (inl |` A ) ` a ) ) )
19	9, 18	sylan 283	. . 3 |- ( ( ph /\ a e. A ) -> ( ( H o. (inl |` A ) ) ` a ) = ( H ` ( (inl |` A ) ` a ) ) )
20		fvres 5714	. . . . . 6 |- ( a e. A -> ( (inl |` A ) ` a ) = (inl ` a ) )
21	20	adantl 277	. . . . 5 |- ( ( ph /\ a e. A ) -> ( (inl |` A ) ` a ) = (inl ` a ) )
22	21	fveq2d 5694	. . . 4 |- ( ( ph /\ a e. A ) -> ( H ` ( (inl |` A ) ` a ) ) = ( H ` (inl ` a ) ) )
23	3	a1i 9	. . . . 5 |- ( ( ph /\ a e. A ) -> H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) ) )
24		fveq2 5690	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 1st ` x ) = ( 1st ` (inl ` a ) ) )
25	24	eqeq1d 2247	. . . . . . . 8 |- ( x = (inl ` a ) -> ( ( 1st ` x ) = (/) <-> ( 1st ` (inl ` a ) ) = (/) ) )
26		fveq2 5690	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 2nd ` x ) = ( 2nd ` (inl ` a ) ) )
27	26	fveq2d 5694	. . . . . . . 8 |- ( x = (inl ` a ) -> ( F ` ( 2nd ` x ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
28	26	fveq2d 5694	. . . . . . . 8 |- ( x = (inl ` a ) -> ( G ` ( 2nd ` x ) ) = ( G ` ( 2nd ` (inl ` a ) ) ) )
29	25, 27, 28	ifbieq12d 3664	. . . . . . 7 |- ( x = (inl ` a ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = if ( ( 1st ` (inl ` a ) ) = (/) , ( F ` ( 2nd ` (inl ` a ) ) ) , ( G ` ( 2nd ` (inl ` a ) ) ) ) )
30	29	adantl 277	. . . . . 6 |- ( ( ( ph /\ a e. A ) /\ x = (inl ` a ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = if ( ( 1st ` (inl ` a ) ) = (/) , ( F ` ( 2nd ` (inl ` a ) ) ) , ( G ` ( 2nd ` (inl ` a ) ) ) ) )
31		1stinl 7404	. . . . . . . . 9 |- ( a e. A -> ( 1st ` (inl ` a ) ) = (/) )
32	31	adantl 277	. . . . . . . 8 |- ( ( ph /\ a e. A ) -> ( 1st ` (inl ` a ) ) = (/) )
33	32	adantr 276	. . . . . . 7 |- ( ( ( ph /\ a e. A ) /\ x = (inl ` a ) ) -> ( 1st ` (inl ` a ) ) = (/) )
34	33	iftrued 3644	. . . . . 6 |- ( ( ( ph /\ a e. A ) /\ x = (inl ` a ) ) -> if ( ( 1st ` (inl ` a ) ) = (/) , ( F ` ( 2nd ` (inl ` a ) ) ) , ( G ` ( 2nd ` (inl ` a ) ) ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
35	30, 34	eqtrd 2271	. . . . 5 |- ( ( ( ph /\ a e. A ) /\ x = (inl ` a ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
36		djulcl 7381	. . . . . 6 |- ( a e. A -> (inl ` a ) e. ( A ⊔ B ) )
37	36	adantl 277	. . . . 5 |- ( ( ph /\ a e. A ) -> (inl ` a ) e. ( A ⊔ B ) )
38	1	adantr 276	. . . . . 6 |- ( ( ph /\ a e. A ) -> F : A --> C )
39		2ndinl 7405	. . . . . . . 8 |- ( a e. A -> ( 2nd ` (inl ` a ) ) = a )
40	39	adantl 277	. . . . . . 7 |- ( ( ph /\ a e. A ) -> ( 2nd ` (inl ` a ) ) = a )
41		simpr 110	. . . . . . 7 |- ( ( ph /\ a e. A ) -> a e. A )
42	40, 41	eqeltrd 2315	. . . . . 6 |- ( ( ph /\ a e. A ) -> ( 2nd ` (inl ` a ) ) e. A )
43	38, 42	ffvelcdmd 5835	. . . . 5 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) e. C )
44	23, 35, 37, 43	fvmptd 5780	. . . 4 |- ( ( ph /\ a e. A ) -> ( H ` (inl ` a ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
45	22, 44	eqtrd 2271	. . 3 |- ( ( ph /\ a e. A ) -> ( H ` ( (inl |` A ) ` a ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
46	40	fveq2d 5694	. . 3 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) = ( F ` a ) )
47	19, 45, 46	3eqtrd 2275	. 2 |- ( ( ph /\ a e. A ) -> ( ( H o. (inl |` A ) ) ` a ) = ( F ` a ) )
48	15, 17, 47	eqfnfvd 5800	1 |- ( ph -> ( H o. (inl |` A ) ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520   ifcif 3635   |-> cmpt 4187   ran crn 4770   |` cres 4771   o. ccom 4773   Fn wfn 5367   -->wf 5368   -1-1->wf1 5369   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   ⊔ cdju 7367  inlcinl 7375

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-dju 7368  df-inl 7377  df-inr 7378

This theorem is referenced by:  updjud  7412