Theorem updjudhcoinlf 7420

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 7419	. . . 4 |- ( ph -> H : ( A ⊔ B ) --> C )
5		ffn 5533	. . . 4 |- ( H : ( A ⊔ B ) --> C -> H Fn ( A ⊔ B ) )
6	4, 5	syl 14	. . 3 |- ( ph -> H Fn ( A ⊔ B ) )
7		inlresf1 7401	. . . 4 |- (inl |` A ) : A -1-1-> ( A ⊔ B )
8		f1fn 5600	. . . 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 5598	. . . . 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 5542	. . . 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 5491	. . 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 5533	. . 3 |- ( F : A --> C -> F Fn A )
17	1, 16	syl 14	. 2 |- ( ph -> F Fn A )
18		fvco2 5774	. . . 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 5719	. . . . . 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 5699	. . . 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 5695	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 1st ` x ) = ( 1st ` (inl ` a ) ) )
25	24	eqeq1d 2247	. . . . . . . 8 |- ( x = (inl ` a ) -> ( ( 1st ` x ) = (/) <-> ( 1st ` (inl ` a ) ) = (/) ) )
26		fveq2 5695	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 2nd ` x ) = ( 2nd ` (inl ` a ) ) )
27	26	fveq2d 5699	. . . . . . . 8 |- ( x = (inl ` a ) -> ( F ` ( 2nd ` x ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
28	26	fveq2d 5699	. . . . . . . 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 7414	. . . . . . . . 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 7391	. . . . . 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 7415	. . . . . . . 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 5844	. . . . 5 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) e. C )
44	23, 35, 37, 43	fvmptd 5786	. . . 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 5699	. . 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 5809	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 4192   ran crn 4775   |` cres 4776   o. ccom 4778   Fn wfn 5372   -->wf 5373   -1-1->wf1 5374   ` cfv 5377   1stc1st 6372   2ndc2nd 6373   ⊔ cdju 7377  inlcinl 7385

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 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375  df-1o 6687  df-dju 7378  df-inl 7387  df-inr 7388

This theorem is used by:  updjud  7422