Theorem updjudhcoinlf 7208

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 7207	. . . 4 |- ( ph -> H : ( A ⊔ B ) --> C )
5		ffn 5445	. . . 4 |- ( H : ( A ⊔ B ) --> C -> H Fn ( A ⊔ B ) )
6	4, 5	syl 14	. . 3 |- ( ph -> H Fn ( A ⊔ B ) )
7		inlresf1 7189	. . . 4 |- (inl |` A ) : A -1-1-> ( A ⊔ B )
8		f1fn 5505	. . . 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 5503	. . . . 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 5454	. . . 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 5403	. . 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 1250	. 2 |- ( ph -> ( H o. (inl |` A ) ) Fn A )
16		ffn 5445	. . 3 |- ( F : A --> C -> F Fn A )
17	1, 16	syl 14	. 2 |- ( ph -> F Fn A )
18		fvco2 5671	. . . 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 5623	. . . . . 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 5603	. . . 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 5599	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 1st ` x ) = ( 1st ` (inl ` a ) ) )
25	24	eqeq1d 2216	. . . . . . . 8 |- ( x = (inl ` a ) -> ( ( 1st ` x ) = (/) <-> ( 1st ` (inl ` a ) ) = (/) ) )
26		fveq2 5599	. . . . . . . . 9 |- ( x = (inl ` a ) -> ( 2nd ` x ) = ( 2nd ` (inl ` a ) ) )
27	26	fveq2d 5603	. . . . . . . 8 |- ( x = (inl ` a ) -> ( F ` ( 2nd ` x ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
28	26	fveq2d 5603	. . . . . . . 8 |- ( x = (inl ` a ) -> ( G ` ( 2nd ` x ) ) = ( G ` ( 2nd ` (inl ` a ) ) ) )
29	25, 27, 28	ifbieq12d 3606	. . . . . . 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 7202	. . . . . . . . 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 3586	. . . . . 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 2240	. . . . 5 |- ( ( ( ph /\ a e. A ) /\ x = (inl ` a ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
36		djulcl 7179	. . . . . 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 7203	. . . . . . . 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 2284	. . . . . 6 |- ( ( ph /\ a e. A ) -> ( 2nd ` (inl ` a ) ) e. A )
43	38, 42	ffvelcdmd 5739	. . . . 5 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) e. C )
44	23, 35, 37, 43	fvmptd 5683	. . . 4 |- ( ( ph /\ a e. A ) -> ( H ` (inl ` a ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
45	22, 44	eqtrd 2240	. . 3 |- ( ( ph /\ a e. A ) -> ( H ` ( (inl |` A ) ` a ) ) = ( F ` ( 2nd ` (inl ` a ) ) ) )
46	40	fveq2d 5603	. . 3 |- ( ( ph /\ a e. A ) -> ( F ` ( 2nd ` (inl ` a ) ) ) = ( F ` a ) )
47	19, 45, 46	3eqtrd 2244	. 2 |- ( ( ph /\ a e. A ) -> ( ( H o. (inl |` A ) ) ` a ) = ( F ` a ) )
48	15, 17, 47	eqfnfvd 5703	1 |- ( ph -> ( H o. (inl |` A ) ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   C_ wss 3174   (/)c0 3468   ifcif 3579   |-> cmpt 4121   ran crn 4694   |` cres 4695   o. ccom 4697   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   ` cfv 5290   1stc1st 6247   2ndc2nd 6248   ⊔ cdju 7165  inlcinl 7173

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-dju 7166  df-inl 7175  df-inr 7176

This theorem is referenced by:  updjud  7210