Theorem updjudhcoinrg 7082

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 7080	. . . 4 |- ( ph -> H : ( A ⊔ B ) --> C )
5		ffn 5367	. . . 4 |- ( H : ( A ⊔ B ) --> C -> H Fn ( A ⊔ B ) )
6	4, 5	syl 14	. . 3 |- ( ph -> H Fn ( A ⊔ B ) )
7		inrresf1 7063	. . . 4 |- (inr |` B ) : B -1-1-> ( A ⊔ B )
8		f1fn 5425	. . . 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 5423	. . . . 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 5376	. . . 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 5326	. . 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 1238	. 2 |- ( ph -> ( H o. (inr |` B ) ) Fn B )
16		ffn 5367	. . 3 |- ( G : B --> C -> G Fn B )
17	2, 16	syl 14	. 2 |- ( ph -> G Fn B )
18		fvco2 5587	. . . 4 |- ( ( (inr |` B ) Fn B /\ b e. B ) -> ( ( H o. (inr |` B ) ) ` b ) = ( H ` ( (inr |` B ) ` b ) ) )
19	9, 18	sylan 283	. . 3 |- ( ( ph /\ b e. B ) -> ( ( H o. (inr |` B ) ) ` b ) = ( H ` ( (inr |` B ) ` b ) ) )
20		fvres 5541	. . . . . 6 |- ( b e. B -> ( (inr |` B ) ` b ) = (inr ` b ) )
21	20	adantl 277	. . . . 5 |- ( ( ph /\ b e. B ) -> ( (inr |` B ) ` b ) = (inr ` b ) )
22	21	fveq2d 5521	. . . 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 5517	. . . . . . . . 9 |- ( x = (inr ` b ) -> ( 1st ` x ) = ( 1st ` (inr ` b ) ) )
25	24	eqeq1d 2186	. . . . . . . 8 |- ( x = (inr ` b ) -> ( ( 1st ` x ) = (/) <-> ( 1st ` (inr ` b ) ) = (/) ) )
26		fveq2 5517	. . . . . . . . 9 |- ( x = (inr ` b ) -> ( 2nd ` x ) = ( 2nd ` (inr ` b ) ) )
27	26	fveq2d 5521	. . . . . . . 8 |- ( x = (inr ` b ) -> ( F ` ( 2nd ` x ) ) = ( F ` ( 2nd ` (inr ` b ) ) ) )
28	26	fveq2d 5521	. . . . . . . 8 |- ( x = (inr ` b ) -> ( G ` ( 2nd ` x ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
29	25, 27, 28	ifbieq12d 3562	. . . . . . 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 277	. . . . . 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 7077	. . . . . . . . . 10 |- ( b e. B -> ( 1st ` (inr ` b ) ) = 1o )
32		1n0 6435	. . . . . . . . . . . 12 |- 1o =/= (/)
33	32	neii 2349	. . . . . . . . . . 11 |- -. 1o = (/)
34		eqeq1 2184	. . . . . . . . . . 11 |- ( ( 1st ` (inr ` b ) ) = 1o -> ( ( 1st ` (inr ` b ) ) = (/) <-> 1o = (/) ) )
35	33, 34	mtbiri 675	. . . . . . . . . 10 |- ( ( 1st ` (inr ` b ) ) = 1o -> -. ( 1st ` (inr ` b ) ) = (/) )
36	31, 35	syl 14	. . . . . . . . 9 |- ( b e. B -> -. ( 1st ` (inr ` b ) ) = (/) )
37	36	adantl 277	. . . . . . . 8 |- ( ( ph /\ b e. B ) -> -. ( 1st ` (inr ` b ) ) = (/) )
38	37	adantr 276	. . . . . . 7 |- ( ( ( ph /\ b e. B ) /\ x = (inr ` b ) ) -> -. ( 1st ` (inr ` b ) ) = (/) )
39	38	iffalsed 3546	. . . . . 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 2210	. . . . 5 |- ( ( ( ph /\ b e. B ) /\ x = (inr ` b ) ) -> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
41		djurcl 7053	. . . . . 6 |- ( b e. B -> (inr ` b ) e. ( A ⊔ B ) )
42	41	adantl 277	. . . . 5 |- ( ( ph /\ b e. B ) -> (inr ` b ) e. ( A ⊔ B ) )
43	2	adantr 276	. . . . . 6 |- ( ( ph /\ b e. B ) -> G : B --> C )
44		2ndinr 7078	. . . . . . . 8 |- ( b e. B -> ( 2nd ` (inr ` b ) ) = b )
45	44	adantl 277	. . . . . . 7 |- ( ( ph /\ b e. B ) -> ( 2nd ` (inr ` b ) ) = b )
46		simpr 110	. . . . . . 7 |- ( ( ph /\ b e. B ) -> b e. B )
47	45, 46	eqeltrd 2254	. . . . . 6 |- ( ( ph /\ b e. B ) -> ( 2nd ` (inr ` b ) ) e. B )
48	43, 47	ffvelcdmd 5654	. . . . 5 |- ( ( ph /\ b e. B ) -> ( G ` ( 2nd ` (inr ` b ) ) ) e. C )
49	23, 40, 42, 48	fvmptd 5599	. . . 4 |- ( ( ph /\ b e. B ) -> ( H ` (inr ` b ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
50	22, 49	eqtrd 2210	. . 3 |- ( ( ph /\ b e. B ) -> ( H ` ( (inr |` B ) ` b ) ) = ( G ` ( 2nd ` (inr ` b ) ) ) )
51	45	fveq2d 5521	. . 3 |- ( ( ph /\ b e. B ) -> ( G ` ( 2nd ` (inr ` b ) ) ) = ( G ` b ) )
52	19, 50, 51	3eqtrd 2214	. 2 |- ( ( ph /\ b e. B ) -> ( ( H o. (inr |` B ) ) ` b ) = ( G ` b ) )
53	15, 17, 52	eqfnfvd 5618	1 |- ( ph -> ( H o. (inr |` B ) ) = G )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   C_ wss 3131   (/)c0 3424   ifcif 3536   |-> cmpt 4066   ran crn 4629   |` cres 4630   o. ccom 4632   Fn wfn 5213   -->wf 5214   -1-1->wf1 5215   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   1oc1o 6412   ⊔ cdju 7038  inrcinr 7047

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by:  updjud  7083