Theorem unpreima 5718

Description: Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
unpreima	|- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )

Proof of Theorem unpreima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 5310	. 2 |- ( Fun F <-> F Fn dom F )
2		elpreima 5712	. . . 4 |- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) ) )
3		elun 3318	. . . . . . 7 |- ( ( F ` x ) e. ( A u. B ) <-> ( ( F ` x ) e. A \/ ( F ` x ) e. B ) )
4	3	anbi2i 457	. . . . . 6 |- ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> ( x e. dom F /\ ( ( F ` x ) e. A \/ ( F ` x ) e. B ) ) )
5		andi 820	. . . . . 6 |- ( ( x e. dom F /\ ( ( F ` x ) e. A \/ ( F ` x ) e. B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) )
6	4, 5	bitri 184	. . . . 5 |- ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) )
7		elun 3318	. . . . . 6 |- ( x e. ( ( `' F " A ) u. ( `' F " B ) ) <-> ( x e. ( `' F " A ) \/ x e. ( `' F " B ) ) )
8		elpreima 5712	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
9		elpreima 5712	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " B ) <-> ( x e. dom F /\ ( F ` x ) e. B ) ) )
10	8, 9	orbi12d 795	. . . . . 6 |- ( F Fn dom F -> ( ( x e. ( `' F " A ) \/ x e. ( `' F " B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) ) )
11	7, 10	bitrid 192	. . . . 5 |- ( F Fn dom F -> ( x e. ( ( `' F " A ) u. ( `' F " B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) ) )
12	6, 11	bitr4id 199	. . . 4 |- ( F Fn dom F -> ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> x e. ( ( `' F " A ) u. ( `' F " B ) ) ) )
13	2, 12	bitrd 188	. . 3 |- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> x e. ( ( `' F " A ) u. ( `' F " B ) ) ) )
14	13	eqrdv 2204	. 2 |- ( F Fn dom F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
15	1, 14	sylbi 121	1 |- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2177   u. cun 3168   `'ccnv 4682   dom cdm 4683   "cima 4686   Fun wfun 5274   Fn wfn 5275   ` cfv 5280

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-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-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288

This theorem is referenced by: (None)