Theorem fnunres1 29417

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

Assertion

Ref	Expression
fnunres1	|- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` A ) = F )

Proof of Theorem fnunres1

Step	Hyp	Ref	Expression
1		fndm 5990	. . . 4 |- ( F Fn A -> dom F = A )
2	1	3ad2ant1 1082	. . 3 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> dom F = A )
3	2	reseq2d 5396	. 2 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` dom F ) = ( ( F u. G ) |` A ) )
4		fnrel 5989	. . . 4 |- ( F Fn A -> Rel F )
5	4	3ad2ant1 1082	. . 3 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> Rel F )
6		fndm 5990	. . . . . 6 |- ( G Fn B -> dom G = B )
7	6	3ad2ant2 1083	. . . . 5 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> dom G = B )
8	2, 7	ineq12d 3815	. . . 4 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( dom F i^i dom G ) = ( A i^i B ) )
9		simp3 1063	. . . 4 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
10	8, 9	eqtrd 2656	. . 3 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( dom F i^i dom G ) = (/) )
11		funresdm1 29416	. . 3 |- ( ( Rel F /\ ( dom F i^i dom G ) = (/) ) -> ( ( F u. G ) |` dom F ) = F )
12	5, 10, 11	syl2anc 693	. 2 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` dom F ) = F )
13	3, 12	eqtr3d 2658	1 |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` A ) = F )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   u. cun 3572   i^i cin 3573   (/)c0 3915   dom cdm 5114   |` cres 5116   Rel wrel 5119   Fn wfn 5883

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  df-fun 5890  df-fn 5891

This theorem is referenced by:  actfunsnf1o  30682