Theorem wunr1om 9541

Description: A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypothesis

Ref	Expression
wun0.1	|- ( ph -> U e. WUni )

Assertion

Ref	Expression
wunr1om	|- ( ph -> ( R1 " om ) C_ U )

Proof of Theorem wunr1om

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( x = (/) -> ( R1 ` x ) = ( R1 ` (/) ) )
2	1	eleq1d 2686	. . . . . 6 |- ( x = (/) -> ( ( R1 ` x ) e. U <-> ( R1 ` (/) ) e. U ) )
3		fveq2 6191	. . . . . . 7 |- ( x = y -> ( R1 ` x ) = ( R1 ` y ) )
4	3	eleq1d 2686	. . . . . 6 |- ( x = y -> ( ( R1 ` x ) e. U <-> ( R1 ` y ) e. U ) )
5		fveq2 6191	. . . . . . 7 |- ( x = suc y -> ( R1 ` x ) = ( R1 ` suc y ) )
6	5	eleq1d 2686	. . . . . 6 |- ( x = suc y -> ( ( R1 ` x ) e. U <-> ( R1 ` suc y ) e. U ) )
7		r10 8631	. . . . . . 7 |- ( R1 ` (/) ) = (/)
8		wun0.1	. . . . . . . 8 |- ( ph -> U e. WUni )
9	8	wun0 9540	. . . . . . 7 |- ( ph -> (/) e. U )
10	7, 9	syl5eqel 2705	. . . . . 6 |- ( ph -> ( R1 ` (/) ) e. U )
11	8	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( R1 ` y ) e. U ) -> U e. WUni )
12		simpr 477	. . . . . . . . 9 |- ( ( ph /\ ( R1 ` y ) e. U ) -> ( R1 ` y ) e. U )
13	11, 12	wunpw 9529	. . . . . . . 8 |- ( ( ph /\ ( R1 ` y ) e. U ) -> ~P ( R1 ` y ) e. U )
14		nnon 7071	. . . . . . . . . 10 |- ( y e. om -> y e. On )
15		r1suc 8633	. . . . . . . . . 10 |- ( y e. On -> ( R1 ` suc y ) = ~P ( R1 ` y ) )
16	14, 15	syl 17	. . . . . . . . 9 |- ( y e. om -> ( R1 ` suc y ) = ~P ( R1 ` y ) )
17	16	eleq1d 2686	. . . . . . . 8 |- ( y e. om -> ( ( R1 ` suc y ) e. U <-> ~P ( R1 ` y ) e. U ) )
18	13, 17	syl5ibr 236	. . . . . . 7 |- ( y e. om -> ( ( ph /\ ( R1 ` y ) e. U ) -> ( R1 ` suc y ) e. U ) )
19	18	expd 452	. . . . . 6 |- ( y e. om -> ( ph -> ( ( R1 ` y ) e. U -> ( R1 ` suc y ) e. U ) ) )
20	2, 4, 6, 10, 19	finds2 7094	. . . . 5 |- ( x e. om -> ( ph -> ( R1 ` x ) e. U ) )
21		eleq1 2689	. . . . . 6 |- ( ( R1 ` x ) = y -> ( ( R1 ` x ) e. U <-> y e. U ) )
22	21	imbi2d 330	. . . . 5 |- ( ( R1 ` x ) = y -> ( ( ph -> ( R1 ` x ) e. U ) <-> ( ph -> y e. U ) ) )
23	20, 22	syl5ibcom 235	. . . 4 |- ( x e. om -> ( ( R1 ` x ) = y -> ( ph -> y e. U ) ) )
24	23	rexlimiv 3027	. . 3 |- ( E. x e. om ( R1 ` x ) = y -> ( ph -> y e. U ) )
25		r1fnon 8630	. . . . 5 |- R1 Fn On
26		fnfun 5988	. . . . 5 |- ( R1 Fn On -> Fun R1 )
27	25, 26	ax-mp 5	. . . 4 |- Fun R1
28		fvelima 6248	. . . 4 |- ( ( Fun R1 /\ y e. ( R1 " om ) ) -> E. x e. om ( R1 ` x ) = y )
29	27, 28	mpan 706	. . 3 |- ( y e. ( R1 " om ) -> E. x e. om ( R1 ` x ) = y )
30	24, 29	syl11 33	. 2 |- ( ph -> ( y e. ( R1 " om ) -> y e. U ) )
31	30	ssrdv 3609	1 |- ( ph -> ( R1 " om ) C_ U )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   "cima 5117   Oncon0 5723   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   omcom 7065   R1cr1 8625  WUnicwun 9522

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-3or 1038  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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-wun 9524

This theorem is referenced by:  wunom  9542