Theorem ovn0ssdmfun 41767

Description: If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6226. (Contributed by AV, 27-Jan-2020.)

Assertion

Ref	Expression
ovn0ssdmfun	|- ( A. a e. D A. b e. E ( a F b ) =/= (/) -> ( ( D X. E ) C_ dom F /\ Fun ( F |` ( D X. E ) ) ) )

Distinct variable groups:   D, a, b   E, a, b   F, a, b

Proof of Theorem ovn0ssdmfun

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( p = <. a , b >. -> ( F ` p ) = ( F ` <. a , b >. ) )
2		df-ov 6653	. . . . 5 |- ( a F b ) = ( F ` <. a , b >. )
3	1, 2	syl6eqr 2674	. . . 4 |- ( p = <. a , b >. -> ( F ` p ) = ( a F b ) )
4	3	neeq1d 2853	. . 3 |- ( p = <. a , b >. -> ( ( F ` p ) =/= (/) <-> ( a F b ) =/= (/) ) )
5	4	ralxp 5263	. 2 |- ( A. p e. ( D X. E ) ( F ` p ) =/= (/) <-> A. a e. D A. b e. E ( a F b ) =/= (/) )
6		fvn0ssdmfun 6350	. 2 |- ( A. p e. ( D X. E ) ( F ` p ) =/= (/) -> ( ( D X. E ) C_ dom F /\ Fun ( F |` ( D X. E ) ) ) )
7	5, 6	sylbir 225	1 |- ( A. a e. D A. b e. E ( a F b ) =/= (/) -> ( ( D X. E ) C_ dom F /\ Fun ( F |` ( D X. E ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   <.cop 4183   X. cxp 5112   dom cdm 5114   |` cres 5116   Fun wfun 5882   ` cfv 5888  (class class class)co 6650

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-8 1992  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-pow 4843  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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)