Theorem updjudhcoinrg 7015

Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second 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
updjudhcoinrg	|- ( ph -> ( H o. (inr |` B ) ) = G )

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

Allowed substitution hint:   H( x)

Proof of Theorem updjudhcoinrg

Dummy variable b 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 7013	. . . 4 |- ( ph -> H : ( A ⊔ B ) --> C )
5		ffn 5316	. . . 4 |- ( H : ( A ⊔ B ) --> C -> H Fn ( A ⊔ B ) )
6	4, 5	syl 14	. . 3 |- ( ph -> H Fn ( A ⊔ B ) )
7		inrresf1 6996	. . . 4 |- (inr |` B ) : B -1-1-> ( A ⊔ B )
8		f1fn 5374	. . . 4 |- ( (inr |` B ) : B -1-1-> ( A ⊔ B ) -> (inr |` B ) Fn B )
9	7, 8	mp1i 10	. . 3 |- ( ph -> (inr |` B ) Fn B )
10		f1f 5372	. . . . 5 |- ( (inr |` B ) : B -1-1-> ( A ⊔ B ) -> (inr |` B ) : B --> ( A ⊔ B ) )
11	7, 10	ax-mp 5	. . . 4 |- (inr |` B ) : B --> ( A ⊔ B )
12		frn 5325	. . . 4 |- ( (inr |` B ) : B --> ( A ⊔ B ) -> ran (inr |` B ) C_ ( A ⊔ B ) )
13	11, 12	mp1i 10	. . 3 |- ( ph -> ran (inr |` B ) C_ ( A ⊔ B ) )
14		fnco 5275	. . 3 |- ( ( H Fn ( A ⊔ B ) /\ (inr |` B ) Fn B /\ ran (inr |` B ) C_ ( A ⊔ B ) ) -> ( H o. (inr |` B ) ) Fn B )
15	6, 9, 13, 14	syl3anc 1220	. 2 |- ( ph -> ( H o. (inr |` B ) ) Fn B )
16		ffn 5316	. . 3 |- ( G : B --> C -> G Fn B )
17	2, 16	syl 14	. 2 |- ( ph -> G Fn B )
18		fvco2 5534	. . . 4 |- ( ( (inr |` B ) Fn B /\ b e. B ) -> ( ( H o. (inr |` B ) ) ` b ) = ( H ` ( (inr |` B ) ` b ) ) )
19	9, 18	sylan 281	. . 3 |- ( ( ph /\ b e. B ) -> ( ( H o. (inr |` B ) ) ` b ) = ( H ` ( (inr |` B ) ` b ) ) )
20		fvres 5489	. . . . . 6 |- ( b e. B -> ( (inr |` B ) ` b ) = (inr ` b ) )
21	20	adantl 275	. . . . 5 |- ( ( ph /\ b e. B ) -> ( (inr |` B ) ` b ) = (inr ` b ) )
22	21	fveq2d 5469	. . . 4 |- ( ( ph /\ b e. B ) -> ( H ` ( (inr |` B ) ` b ) ) = ( H ` (inr ` b ) ) )
23	3	a1i 9	. . . . 5 |- ( ( ph /\ b e. B ) -> H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) ) )
24		fveq2 5465	. . . . . . . . 9 |- ( x = (inr ` b ) -> ( 1st ` x ) = ( 1st ` (inr ` b ) ) )
25	24	eqeq1d 2166	. . . . . . . 8 |- ( x = (inr ` b ) -> ( ( 1st ` x ) = (/) <-> ( 1st ` (inr ` b ) ) = (/) ) )
26		fveq2 5465	. . . . . . . . 9 |- ( x = (inr ` b ) -> ( 2nd ` x ) = ( 2nd ` (inr ` b ) ) )
27	26	fveq2d 5469	. . . . . . . 8 |- ( x = (inr ` b ) -> ( F ` ( 2nd ` x ) ) = ( F ` ( 2nd ` (inr ` b ) ) ) )
28	26	fveq2d 5469	. . . . . . . 8 |- ( x = (inr ` b ) -> ( G ` ( 2nd ` x ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
29	25, 27, 28	ifbieq12d 3531	. . . . . . 7 |- ( x = (inr ` b ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = if ( ( 1st ` (inr ` b ) ) = (/) , ( F ` ( 2nd ` (inr ` b ) ) ) , ( G ` ( 2nd ` (inr ` b ) ) ) ) )
30	29	adantl 275	. . . . . 6 |- ( ( ( ph /\ b e. B ) /\ x = (inr ` b ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = if ( ( 1st ` (inr ` b ) ) = (/) , ( F ` ( 2nd ` (inr ` b ) ) ) , ( G ` ( 2nd ` (inr ` b ) ) ) ) )
31		1stinr 7010	. . . . . . . . . 10 |- ( b e. B -> ( 1st ` (inr ` b ) ) = 1o )
32		1n0 6373	. . . . . . . . . . . 12 |- 1o =/= (/)
33	32	neii 2329	. . . . . . . . . . 11 |- -. 1o = (/)
34		eqeq1 2164	. . . . . . . . . . 11 |- ( ( 1st ` (inr ` b ) ) = 1o -> ( ( 1st ` (inr ` b ) ) = (/) <-> 1o = (/) ) )
35	33, 34	mtbiri 665	. . . . . . . . . 10 |- ( ( 1st ` (inr ` b ) ) = 1o -> -. ( 1st ` (inr ` b ) ) = (/) )
36	31, 35	syl 14	. . . . . . . . 9 |- ( b e. B -> -. ( 1st ` (inr ` b ) ) = (/) )
37	36	adantl 275	. . . . . . . 8 |- ( ( ph /\ b e. B ) -> -. ( 1st ` (inr ` b ) ) = (/) )
38	37	adantr 274	. . . . . . 7 |- ( ( ( ph /\ b e. B ) /\ x = (inr ` b ) ) -> -. ( 1st ` (inr ` b ) ) = (/) )
39	38	iffalsed 3515	. . . . . 6 |- ( ( ( ph /\ b e. B ) /\ x = (inr ` b ) ) -> if ( ( 1st ` (inr ` b ) ) = (/) , ( F ` ( 2nd ` (inr ` b ) ) ) , ( G ` ( 2nd ` (inr ` b ) ) ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
40	30, 39	eqtrd 2190	. . . . 5 |- ( ( ( ph /\ b e. B ) /\ x = (inr ` b ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
41		djurcl 6986	. . . . . 6 |- ( b e. B -> (inr ` b ) e. ( A ⊔ B ) )
42	41	adantl 275	. . . . 5 |- ( ( ph /\ b e. B ) -> (inr ` b ) e. ( A ⊔ B ) )
43	2	adantr 274	. . . . . 6 |- ( ( ph /\ b e. B ) -> G : B --> C )
44		2ndinr 7011	. . . . . . . 8 |- ( b e. B -> ( 2nd ` (inr ` b ) ) = b )
45	44	adantl 275	. . . . . . 7 |- ( ( ph /\ b e. B ) -> ( 2nd ` (inr ` b ) ) = b )
46		simpr 109	. . . . . . 7 |- ( ( ph /\ b e. B ) -> b e. B )
47	45, 46	eqeltrd 2234	. . . . . 6 |- ( ( ph /\ b e. B ) -> ( 2nd ` (inr ` b ) ) e. B )
48	43, 47	ffvelrnd 5600	. . . . 5 |- ( ( ph /\ b e. B ) -> ( G ` ( 2nd ` (inr ` b ) ) ) e. C )
49	23, 40, 42, 48	fvmptd 5546	. . . 4 |- ( ( ph /\ b e. B ) -> ( H ` (inr ` b ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
50	22, 49	eqtrd 2190	. . 3 |- ( ( ph /\ b e. B ) -> ( H ` ( (inr |` B ) ` b ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
51	45	fveq2d 5469	. . 3 |- ( ( ph /\ b e. B ) -> ( G ` ( 2nd ` (inr ` b ) ) ) = ( G ` b ) )
52	19, 50, 51	3eqtrd 2194	. 2 |- ( ( ph /\ b e. B ) -> ( ( H o. (inr |` B ) ) ` b ) = ( G ` b ) )
53	15, 17, 52	eqfnfvd 5565	1 |- ( ph -> ( H o. (inr |` B ) ) = G )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   C_ wss 3102   (/)c0 3394   ifcif 3505   |-> cmpt 4025   ran crn 4584   |` cres 4585   o. ccom 4587   Fn wfn 5162   -->wf 5163   -1-1->wf1 5164   ` cfv 5167   1stc1st 6080   2ndc2nd 6081   1oc1o 6350   ⊔ cdju 6971  inrcinr 6980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-1st 6082  df-2nd 6083  df-1o 6357  df-dju 6972  df-inl 6981  df-inr 6982

This theorem is referenced by:  updjud  7016