Theorem fveqdmss 6354

Description: If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.)

Hypothesis

Ref	Expression
fveqdmss.1	|- D = dom B

Assertion

Ref	Expression
fveqdmss	|- ( ( Fun B /\ (/) e/ ran B /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> D C_ dom A )

Distinct variable groups:   x, A   x, B   x, D

Proof of Theorem fveqdmss

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . . 9 |- ( x = a -> ( A ` x ) = ( A ` a ) )
2		fveq2 6191	. . . . . . . . 9 |- ( x = a -> ( B ` x ) = ( B ` a ) )
3	1, 2	eqeq12d 2637	. . . . . . . 8 |- ( x = a -> ( ( A ` x ) = ( B ` x ) <-> ( A ` a ) = ( B ` a ) ) )
4	3	rspcva 3307	. . . . . . 7 |- ( ( a e. D /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> ( A ` a ) = ( B ` a ) )
5		nelrnfvne 6353	. . . . . . . . . . . . 13 |- ( ( Fun B /\ a e. dom B /\ (/) e/ ran B ) -> ( B ` a ) =/= (/) )
6		n0 3931	. . . . . . . . . . . . . 14 |- ( ( B ` a ) =/= (/) <-> E. b b e. ( B ` a ) )
7		eleq2 2690	. . . . . . . . . . . . . . . . . 18 |- ( ( B ` a ) = ( A ` a ) -> ( b e. ( B ` a ) <-> b e. ( A ` a ) ) )
8	7	eqcoms 2630	. . . . . . . . . . . . . . . . 17 |- ( ( A ` a ) = ( B ` a ) -> ( b e. ( B ` a ) <-> b e. ( A ` a ) ) )
9		elfvdm 6220	. . . . . . . . . . . . . . . . 17 |- ( b e. ( A ` a ) -> a e. dom A )
10	8, 9	syl6bi 243	. . . . . . . . . . . . . . . 16 |- ( ( A ` a ) = ( B ` a ) -> ( b e. ( B ` a ) -> a e. dom A ) )
11	10	com12 32	. . . . . . . . . . . . . . 15 |- ( b e. ( B ` a ) -> ( ( A ` a ) = ( B ` a ) -> a e. dom A ) )
12	11	exlimiv 1858	. . . . . . . . . . . . . 14 |- ( E. b b e. ( B ` a ) -> ( ( A ` a ) = ( B ` a ) -> a e. dom A ) )
13	6, 12	sylbi 207	. . . . . . . . . . . . 13 |- ( ( B ` a ) =/= (/) -> ( ( A ` a ) = ( B ` a ) -> a e. dom A ) )
14	5, 13	syl 17	. . . . . . . . . . . 12 |- ( ( Fun B /\ a e. dom B /\ (/) e/ ran B ) -> ( ( A ` a ) = ( B ` a ) -> a e. dom A ) )
15	14	3exp 1264	. . . . . . . . . . 11 |- ( Fun B -> ( a e. dom B -> ( (/) e/ ran B -> ( ( A ` a ) = ( B ` a ) -> a e. dom A ) ) ) )
16	15	com12 32	. . . . . . . . . 10 |- ( a e. dom B -> ( Fun B -> ( (/) e/ ran B -> ( ( A ` a ) = ( B ` a ) -> a e. dom A ) ) ) )
17		fveqdmss.1	. . . . . . . . . 10 |- D = dom B
18	16, 17	eleq2s 2719	. . . . . . . . 9 |- ( a e. D -> ( Fun B -> ( (/) e/ ran B -> ( ( A ` a ) = ( B ` a ) -> a e. dom A ) ) ) )
19	18	com24 95	. . . . . . . 8 |- ( a e. D -> ( ( A ` a ) = ( B ` a ) -> ( (/) e/ ran B -> ( Fun B -> a e. dom A ) ) ) )
20	19	adantr 481	. . . . . . 7 |- ( ( a e. D /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> ( ( A ` a ) = ( B ` a ) -> ( (/) e/ ran B -> ( Fun B -> a e. dom A ) ) ) )
21	4, 20	mpd 15	. . . . . 6 |- ( ( a e. D /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> ( (/) e/ ran B -> ( Fun B -> a e. dom A ) ) )
22	21	ex 450	. . . . 5 |- ( a e. D -> ( A. x e. D ( A ` x ) = ( B ` x ) -> ( (/) e/ ran B -> ( Fun B -> a e. dom A ) ) ) )
23	22	com23 86	. . . 4 |- ( a e. D -> ( (/) e/ ran B -> ( A. x e. D ( A ` x ) = ( B ` x ) -> ( Fun B -> a e. dom A ) ) ) )
24	23	com14 96	. . 3 |- ( Fun B -> ( (/) e/ ran B -> ( A. x e. D ( A ` x ) = ( B ` x ) -> ( a e. D -> a e. dom A ) ) ) )
25	24	3imp 1256	. 2 |- ( ( Fun B /\ (/) e/ ran B /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> ( a e. D -> a e. dom A ) )
26	25	ssrdv 3609	1 |- ( ( Fun B /\ (/) e/ ran B /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> D C_ dom A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   C_ wss 3574   (/)c0 3915   dom cdm 5114   ran crn 5115   Fun wfun 5882   ` cfv 5888

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-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fveqressseq  6355