Theorem tz7.48-2 7537

Description: Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)

Hypothesis

Ref	Expression
tz7.48.1	|- F Fn On

Assertion

Ref	Expression
tz7.48-2	|- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> Fun `' F )

Distinct variable group:   x, F

Allowed substitution hint:   A( x)

Proof of Theorem tz7.48-2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3624	. . 3 |- On C_ On
2		onelon 5748	. . . . . . . . 9 |- ( ( x e. On /\ y e. x ) -> y e. On )
3	2	ancoms 469	. . . . . . . 8 |- ( ( y e. x /\ x e. On ) -> y e. On )
4		tz7.48.1	. . . . . . . . . . 11 |- F Fn On
5		fndm 5990	. . . . . . . . . . 11 |- ( F Fn On -> dom F = On )
6	4, 5	ax-mp 5	. . . . . . . . . 10 |- dom F = On
7	6	eleq2i 2693	. . . . . . . . 9 |- ( y e. dom F <-> y e. On )
8		fnfun 5988	. . . . . . . . . . . . 13 |- ( F Fn On -> Fun F )
9	4, 8	ax-mp 5	. . . . . . . . . . . 12 |- Fun F
10		funfvima 6492	. . . . . . . . . . . 12 |- ( ( Fun F /\ y e. dom F ) -> ( y e. x -> ( F ` y ) e. ( F " x ) ) )
11	9, 10	mpan 706	. . . . . . . . . . 11 |- ( y e. dom F -> ( y e. x -> ( F ` y ) e. ( F " x ) ) )
12	11	impcom 446	. . . . . . . . . 10 |- ( ( y e. x /\ y e. dom F ) -> ( F ` y ) e. ( F " x ) )
13		eleq1a 2696	. . . . . . . . . . 11 |- ( ( F ` y ) e. ( F " x ) -> ( ( F ` x ) = ( F ` y ) -> ( F ` x ) e. ( F " x ) ) )
14		eldifn 3733	. . . . . . . . . . 11 |- ( ( F ` x ) e. ( A \ ( F " x ) ) -> -. ( F ` x ) e. ( F " x ) )
15	13, 14	nsyli 155	. . . . . . . . . 10 |- ( ( F ` y ) e. ( F " x ) -> ( ( F ` x ) e. ( A \ ( F " x ) ) -> -. ( F ` x ) = ( F ` y ) ) )
16	12, 15	syl 17	. . . . . . . . 9 |- ( ( y e. x /\ y e. dom F ) -> ( ( F ` x ) e. ( A \ ( F " x ) ) -> -. ( F ` x ) = ( F ` y ) ) )
17	7, 16	sylan2br 493	. . . . . . . 8 |- ( ( y e. x /\ y e. On ) -> ( ( F ` x ) e. ( A \ ( F " x ) ) -> -. ( F ` x ) = ( F ` y ) ) )
18	3, 17	syldan 487	. . . . . . 7 |- ( ( y e. x /\ x e. On ) -> ( ( F ` x ) e. ( A \ ( F " x ) ) -> -. ( F ` x ) = ( F ` y ) ) )
19	18	expimpd 629	. . . . . 6 |- ( y e. x -> ( ( x e. On /\ ( F ` x ) e. ( A \ ( F " x ) ) ) -> -. ( F ` x ) = ( F ` y ) ) )
20	19	com12 32	. . . . 5 |- ( ( x e. On /\ ( F ` x ) e. ( A \ ( F " x ) ) ) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) )
21	20	ralrimiv 2965	. . . 4 |- ( ( x e. On /\ ( F ` x ) e. ( A \ ( F " x ) ) ) -> A. y e. x -. ( F ` x ) = ( F ` y ) )
22	21	ralimiaa 2951	. . 3 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) )
23	4	tz7.48lem 7536	. . 3 |- ( ( On C_ On /\ A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) ) -> Fun `' ( F |` On ) )
24	1, 22, 23	sylancr 695	. 2 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> Fun `' ( F |` On ) )
25		fnrel 5989	. . . . . 6 |- ( F Fn On -> Rel F )
26	4, 25	ax-mp 5	. . . . 5 |- Rel F
27	6	eqimssi 3659	. . . . 5 |- dom F C_ On
28		relssres 5437	. . . . 5 |- ( ( Rel F /\ dom F C_ On ) -> ( F |` On ) = F )
29	26, 27, 28	mp2an 708	. . . 4 |- ( F |` On ) = F
30	29	cnveqi 5297	. . 3 |- `' ( F |` On ) = `' F
31	30	funeqi 5909	. 2 |- ( Fun `' ( F |` On ) <-> Fun `' F )
32	24, 31	sylib 208	1 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> Fun `' F )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   Rel wrel 5119   Oncon0 5723   Fun wfun 5882   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-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

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-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896

This theorem is referenced by:  tz7.48-3  7539