Theorem funresdm1 29416

Description: Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.)

Assertion

Ref	Expression
funresdm1	|- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> ( ( A u. B ) |` dom A ) = A )

Proof of Theorem funresdm1

Step	Hyp	Ref	Expression
1		resundir 5411	. 2 |- ( ( A u. B ) |` dom A ) = ( ( A |` dom A ) u. ( B |` dom A ) )
2		resdm 5441	. . . . 5 |- ( Rel A -> ( A |` dom A ) = A )
3	2	adantr 481	. . . 4 |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> ( A |` dom A ) = A )
4		dmres 5419	. . . . . 6 |- dom ( B |` dom A ) = ( dom A i^i dom B )
5		simpr 477	. . . . . 6 |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> ( dom A i^i dom B ) = (/) )
6	4, 5	syl5eq 2668	. . . . 5 |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> dom ( B |` dom A ) = (/) )
7		relres 5426	. . . . . 6 |- Rel ( B |` dom A )
8		reldm0 5343	. . . . . 6 |- ( Rel ( B |` dom A ) -> ( ( B |` dom A ) = (/) <-> dom ( B |` dom A ) = (/) ) )
9	7, 8	ax-mp 5	. . . . 5 |- ( ( B |` dom A ) = (/) <-> dom ( B |` dom A ) = (/) )
10	6, 9	sylibr 224	. . . 4 |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> ( B |` dom A ) = (/) )
11	3, 10	uneq12d 3768	. . 3 |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> ( ( A |` dom A ) u. ( B |` dom A ) ) = ( A u. (/) ) )
12		un0 3967	. . 3 |- ( A u. (/) ) = A
13	11, 12	syl6eq 2672	. 2 |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> ( ( A |` dom A ) u. ( B |` dom A ) ) = A )
14	1, 13	syl5eq 2668	1 |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) ) -> ( ( A u. B ) |` dom A ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   u. cun 3572   i^i cin 3573   (/)c0 3915   dom cdm 5114   |` cres 5116   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-res 5126

This theorem is referenced by:  fnunres1  29417