Theorem updjudhcoinlf 7426

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 7425	. . . 4 |- ( ph -> H : ( A ⊔ B ) --> C )
5		ffn 5536	. . . 4 |- ( H : ( A ⊔ B ) --> C -> H Fn ( A ⊔ B ) )
6	4, 5	syl 14	. . 3 |- ( ph -> H Fn ( A ⊔ B ) )
7		inlresf1 7407	. . . 4 |- (inl |` A ) : A -1-1-> ( A ⊔ B )
8		f1fn 5603	. . . 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 5601	. . . . 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 5545	. . . 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 5494	. . 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 5536	. . 3 |- ( F : A --> C -> F Fn A )
17	1, 16	syl 14	. 2 |- ( ph -> F Fn A )
18		fvco2 5777	. . . 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 5722	. . . . . 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 5702	. . . 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 5698	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 1st ` x ) = ( 1st ` (inl ` a ) ) )
25	24	eqeq1d 2247	. . . . . . . 8 |- ( x = (inl ` a ) -> ( ( 1st ` x ) = (/) <-> ( 1st ` (inl ` a ) ) = (/) ) )
26		fveq2 5698	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 2nd ` x ) = ( 2nd ` (inl ` a ) ) )
27	26	fveq2d 5702	. . . . . . . 8 |- ( x = (inl ` a ) -> ( F ` ( 2nd ` x ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
28	26	fveq2d 5702	. . . . . . . 8 |- ( x = (inl ` a ) -> ( G ` ( 2nd ` x ) ) = ( G ` ( 2nd ` (inl ` a ) ) ) )
29	25, 27, 28	ifbieq12d 3667	. . . . . . 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 7420	. . . . . . . . 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 3647	. . . . . 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 7397	. . . . . 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 7421	. . . . . . . 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 5847	. . . . 5 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) e. C )
44	23, 35, 37, 43	fvmptd 5789	. . . 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 5702	. . 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 5812	1 |- ( ph -> ( H o. (inl |` A ) ) = F )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520   ifcif 3638   |-> cmpt 4193   ran crn 4777   |` cres 4778   o. ccom 4780   Fn wfn 5375   -->wf 5376   -1-1->wf1 5377   ` cfv 5380   1stc1st 6376   2ndc2nd 6377   ⊔ cdju 7383  inlcinl 7391

This proof depends on 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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-dju 7384  df-inl 7393  df-inr 7394

This theorem is used by:  updjud  7428