Theorem mpt2xopx0ov0 6994

Description: If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Hypothesis

Ref	Expression
mpt2xopn0yelv.f	|- F = ( x e. _V , y e. ( 1st ` x ) |-> C )

Assertion

Ref	Expression
mpt2xopx0ov0	|- ( (/) F K ) = (/)

Distinct variable groups:   x, y   x, K   x, F

Allowed substitution hints:   C( x, y)   F( y)   K( y)

Proof of Theorem mpt2xopx0ov0

Step	Hyp	Ref	Expression
1		0nelxp 4884	. 2 |- -. (/) e. ( _V X. _V )
2		mpt2xopn0yelv.f	. . 3 |- F = ( x e. _V , y e. ( 1st ` x ) |-> C )
3	2	mpt2xopxnop0 6993	. 2 |- ( -. (/) e. ( _V X. _V ) -> ( (/) F K ) = (/) )
4	1, 3	ax-mp 5	1 |- ( (/) F K ) = (/)

Syntax hints:   -. wn 3   = wceq 1455   e. wcel 1898   _Vcvv 3057   (/)c0 3743   X. cxp 4854   ` cfv 5605  (class class class)co 6320   |-> cmpt2 6322   1stc1st 6823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-1st 6825  df-2nd 6826

This theorem is referenced by: (None)