Theorem updjudhcoinlf 7384

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 7383	. . . 4 |- ( ph -> H : ( A ⊔ B ) --> C )
5		ffn 5513	. . . 4 |- ( H : ( A ⊔ B ) --> C -> H Fn ( A ⊔ B ) )
6	4, 5	syl 14	. . 3 |- ( ph -> H Fn ( A ⊔ B ) )
7		inlresf1 7365	. . . 4 |- (inl |` A ) : A -1-1-> ( A ⊔ B )
8		f1fn 5580	. . . 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 5578	. . . . 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 5522	. . . 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 5471	. . 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 1274	. 2 |- ( ph -> ( H o. (inl |` A ) ) Fn A )
16		ffn 5513	. . 3 |- ( F : A --> C -> F Fn A )
17	1, 16	syl 14	. 2 |- ( ph -> F Fn A )
18		fvco2 5751	. . . 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 5699	. . . . . 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 5679	. . . 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 5675	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 1st ` x ) = ( 1st ` (inl ` a ) ) )
25	24	eqeq1d 2243	. . . . . . . 8 |- ( x = (inl ` a ) -> ( ( 1st ` x ) = (/) <-> ( 1st ` (inl ` a ) ) = (/) ) )
26		fveq2 5675	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 2nd ` x ) = ( 2nd ` (inl ` a ) ) )
27	26	fveq2d 5679	. . . . . . . 8 |- ( x = (inl ` a ) -> ( F ` ( 2nd ` x ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
28	26	fveq2d 5679	. . . . . . . 8 |- ( x = (inl ` a ) -> ( G ` ( 2nd ` x ) ) = ( G ` ( 2nd ` (inl ` a ) ) ) )
29	25, 27, 28	ifbieq12d 3653	. . . . . . 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 7378	. . . . . . . . 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 3633	. . . . . 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 2267	. . . . 5 |- ( ( ( ph /\ a e. A ) /\ x = (inl ` a ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
36		djulcl 7355	. . . . . 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 7379	. . . . . . . 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 2311	. . . . . 6 |- ( ( ph /\ a e. A ) -> ( 2nd ` (inl ` a ) ) e. A )
43	38, 42	ffvelcdmd 5818	. . . . 5 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) e. C )
44	23, 35, 37, 43	fvmptd 5763	. . . 4 |- ( ( ph /\ a e. A ) -> ( H ` (inl ` a ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
45	22, 44	eqtrd 2267	. . 3 |- ( ( ph /\ a e. A ) -> ( H ` ( (inl |` A ) ` a ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
46	40	fveq2d 5679	. . 3 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) = ( F ` a ) )
47	19, 45, 46	3eqtrd 2271	. 2 |- ( ( ph /\ a e. A ) -> ( ( H o. (inl |` A ) ) ` a ) = ( F ` a ) )
48	15, 17, 47	eqfnfvd 5783	1 |- ( ph -> ( H o. (inl |` A ) ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3214   (/)c0 3512   ifcif 3624   |-> cmpt 4176   ran crn 4755   |` cres 4756   o. ccom 4758   Fn wfn 5352   -->wf 5353   -1-1->wf1 5354   ` cfv 5357   1stc1st 6345   2ndc2nd 6346   ⊔ cdju 7341  inlcinl 7349

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-dju 7342  df-inl 7351  df-inr 7352

This theorem is referenced by:  updjud  7386