Theorem tz9.12lem1 8650

Description: Lemma for tz9.12 8653. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
tz9.12lem.1	|- A e. _V
tz9.12lem.2	|- F = ( z e. _V |-> |^| { v e. On | z e. ( R1 ` v ) } )

Assertion

Ref	Expression
tz9.12lem1	|- ( F " A ) C_ On

Distinct variable group:   z, v, A

Allowed substitution hints:   F( z, v)

Proof of Theorem tz9.12lem1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 5477	. 2 |- ( F " A ) C_ ran F
2		tz9.12lem.2	. . . 4 |- F = ( z e. _V |-> |^| { v e. On | z e. ( R1 ` v ) } )
3	2	rnmpt 5371	. . 3 |- ran F = { x | E. z e. _V x = |^| { v e. On | z e. ( R1 ` v ) } }
4		id 22	. . . . . 6 |- ( x = |^| { v e. On | z e. ( R1 ` v ) } -> x = |^| { v e. On | z e. ( R1 ` v ) } )
5		ssrab2 3687	. . . . . . 7 |- { v e. On | z e. ( R1 ` v ) } C_ On
6		eqvisset 3211	. . . . . . . 8 |- ( x = |^| { v e. On | z e. ( R1 ` v ) } -> |^| { v e. On | z e. ( R1 ` v ) } e. _V )
7		intex 4820	. . . . . . . 8 |- ( { v e. On | z e. ( R1 ` v ) } =/= (/) <-> |^| { v e. On | z e. ( R1 ` v ) } e. _V )
8	6, 7	sylibr 224	. . . . . . 7 |- ( x = |^| { v e. On | z e. ( R1 ` v ) } -> { v e. On | z e. ( R1 ` v ) } =/= (/) )
9		oninton 7000	. . . . . . 7 |- ( ( { v e. On | z e. ( R1 ` v ) } C_ On /\ { v e. On | z e. ( R1 ` v ) } =/= (/) ) -> |^| { v e. On | z e. ( R1 ` v ) } e. On )
10	5, 8, 9	sylancr 695	. . . . . 6 |- ( x = |^| { v e. On | z e. ( R1 ` v ) } -> |^| { v e. On | z e. ( R1 ` v ) } e. On )
11	4, 10	eqeltrd 2701	. . . . 5 |- ( x = |^| { v e. On | z e. ( R1 ` v ) } -> x e. On )
12	11	rexlimivw 3029	. . . 4 |- ( E. z e. _V x = |^| { v e. On | z e. ( R1 ` v ) } -> x e. On )
13	12	abssi 3677	. . 3 |- { x | E. z e. _V x = |^| { v e. On | z e. ( R1 ` v ) } } C_ On
14	3, 13	eqsstri 3635	. 2 |- ran F C_ On
15	1, 14	sstri 3612	1 |- ( F " A ) C_ On

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |^|cint 4475   |-> cmpt 4729   ran crn 5115   "cima 5117   Oncon0 5723   ` cfv 5888   R1cr1 8625

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-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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727

This theorem is referenced by:  tz9.12lem2  8651  tz9.12lem3  8652