Theorem 0setrec 42447

Description: If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.)

Hypothesis

Ref	Expression
0setrec.1	|- ( ph -> ( F ` (/) ) = (/) )

Assertion

Ref	Expression
0setrec	|- ( ph -> setrecs ( F ) = (/) )

Proof of Theorem 0setrec

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- setrecs ( F ) = setrecs ( F )
2		ss0 3974	. . . . 5 |- ( x C_ (/) -> x = (/) )
3		fveq2 6191	. . . . . . 7 |- ( x = (/) -> ( F ` x ) = ( F ` (/) ) )
4		0setrec.1	. . . . . . 7 |- ( ph -> ( F ` (/) ) = (/) )
5	3, 4	sylan9eqr 2678	. . . . . 6 |- ( ( ph /\ x = (/) ) -> ( F ` x ) = (/) )
6	5	ex 450	. . . . 5 |- ( ph -> ( x = (/) -> ( F ` x ) = (/) ) )
7		eqimss 3657	. . . . 5 |- ( ( F ` x ) = (/) -> ( F ` x ) C_ (/) )
8	2, 6, 7	syl56 36	. . . 4 |- ( ph -> ( x C_ (/) -> ( F ` x ) C_ (/) ) )
9	8	alrimiv 1855	. . 3 |- ( ph -> A. x ( x C_ (/) -> ( F ` x ) C_ (/) ) )
10	1, 9	setrec2v 42443	. 2 |- ( ph -> setrecs ( F ) C_ (/) )
11		ss0 3974	. 2 |- (setrecs ( F ) C_ (/) -> setrecs ( F ) = (/) )
12	10, 11	syl 17	1 |- ( ph -> setrecs ( F ) = (/) )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   (/)c0 3915   ` cfv 5888  setrecscsetrecs 42430

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-setrecs 42431

This theorem is referenced by: (None)