Theorem tz7.48-1 7538

Description: Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)

Hypothesis

Ref	Expression
tz7.48.1	|- F Fn On

Assertion

Ref	Expression
tz7.48-1	|- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ran F C_ A )

Distinct variable groups:   x, F   x, A

Proof of Theorem tz7.48-1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- y e. _V
2	1	elrn2 5365	. . . 4 |- ( y e. ran F <-> E. x <. x , y >. e. F )
3		vex 3203	. . . . . . . . 9 |- x e. _V
4	3, 1	opeldm 5328	. . . . . . . 8 |- ( <. x , y >. e. F -> x e. dom F )
5		tz7.48.1	. . . . . . . . 9 |- F Fn On
6		fndm 5990	. . . . . . . . 9 |- ( F Fn On -> dom F = On )
7	5, 6	ax-mp 5	. . . . . . . 8 |- dom F = On
8	4, 7	syl6eleq 2711	. . . . . . 7 |- ( <. x , y >. e. F -> x e. On )
9	8	ancri 575	. . . . . 6 |- ( <. x , y >. e. F -> ( x e. On /\ <. x , y >. e. F ) )
10		fnopfvb 6237	. . . . . . . 8 |- ( ( F Fn On /\ x e. On ) -> ( ( F ` x ) = y <-> <. x , y >. e. F ) )
11	5, 10	mpan 706	. . . . . . 7 |- ( x e. On -> ( ( F ` x ) = y <-> <. x , y >. e. F ) )
12	11	pm5.32i 669	. . . . . 6 |- ( ( x e. On /\ ( F ` x ) = y ) <-> ( x e. On /\ <. x , y >. e. F ) )
13	9, 12	sylibr 224	. . . . 5 |- ( <. x , y >. e. F -> ( x e. On /\ ( F ` x ) = y ) )
14	13	eximi 1762	. . . 4 |- ( E. x <. x , y >. e. F -> E. x ( x e. On /\ ( F ` x ) = y ) )
15	2, 14	sylbi 207	. . 3 |- ( y e. ran F -> E. x ( x e. On /\ ( F ` x ) = y ) )
16		nfra1 2941	. . . 4 |- F/ x A. x e. On ( F ` x ) e. ( A \ ( F " x ) )
17		nfv 1843	. . . 4 |- F/ x y e. A
18		rsp 2929	. . . . 5 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ( x e. On -> ( F ` x ) e. ( A \ ( F " x ) ) ) )
19		eldifi 3732	. . . . . . . 8 |- ( ( F ` x ) e. ( A \ ( F " x ) ) -> ( F ` x ) e. A )
20		eleq1 2689	. . . . . . . 8 |- ( ( F ` x ) = y -> ( ( F ` x ) e. A <-> y e. A ) )
21	19, 20	syl5ibcom 235	. . . . . . 7 |- ( ( F ` x ) e. ( A \ ( F " x ) ) -> ( ( F ` x ) = y -> y e. A ) )
22	21	imim2i 16	. . . . . 6 |- ( ( x e. On -> ( F ` x ) e. ( A \ ( F " x ) ) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) )
23	22	impd 447	. . . . 5 |- ( ( x e. On -> ( F ` x ) e. ( A \ ( F " x ) ) ) -> ( ( x e. On /\ ( F ` x ) = y ) -> y e. A ) )
24	18, 23	syl 17	. . . 4 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ( ( x e. On /\ ( F ` x ) = y ) -> y e. A ) )
25	16, 17, 24	exlimd 2087	. . 3 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ( E. x ( x e. On /\ ( F ` x ) = y ) -> y e. A ) )
26	15, 25	syl5 34	. 2 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ( y e. ran F -> y e. A ) )
27	26	ssrdv 3609	1 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ran F C_ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   <.cop 4183   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   Fn wfn 5883   ` 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-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

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-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:  tz7.48-3  7539