Theorem tz7.49c 3945

Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51.

Hypotheses

Ref	Expression
tz7.48.1	|- F Fn On
tz7.49.2	|- A e. V

Assertion

Ref	Expression
tz7.49c	|- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (F |` x):x-1-1-onto->A)

Distinct variable groups:   x,F   x,A

Proof of Theorem tz7.49c

Step	Hyp	Ref	Expression
1		tz7.48.1	. . 3 |- F Fn On
2		tz7.49.2	. . 3 |- A e. V
3	1, 2	tz7.49 3944	. 2 |- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (A.y e. x (A \ (F"y)) =/= (/) /\ (F"x) = A /\ Fun `'(F |` x)))
4		onsst 2982	. . . . . . . . . 10 |- (x e. On -> x (_ On)
5		fnssres 3586	. . . . . . . . . . 11 |- ((F Fn On /\ x (_ On) -> (F |` x) Fn x)
6	1, 5	mpan 693	. . . . . . . . . 10 |- (x (_ On -> (F |` x) Fn x)
7	4, 6	syl 10	. . . . . . . . 9 |- (x e. On -> (F |` x) Fn x)
8		df-ima 3181	. . . . . . . . . . 11 |- (F"x) = ran ( F |` x)
9	8	eqeq1i 1474	. . . . . . . . . 10 |- ((F"x) = A <-> ran ( F |` x) = A)
10	9	biimp 151	. . . . . . . . 9 |- ((F"x) = A -> ran ( F |` x) = A)
11	7, 10	anim12i 333	. . . . . . . 8 |- ((x e. On /\ (F"x) = A) -> ((F |` x) Fn x /\ ran ( F |` x) = A))
12	11	anim1i 334	. . . . . . 7 |- (((x e. On /\ (F"x) = A) /\ Fun `'(F |` x)) -> (((F |` x) Fn x /\ ran ( F |` x) = A) /\ Fun `'(F |` x)))
13		f1o2 3678	. . . . . . . 8 |- ((F |` x):x-1-1-onto->A <-> ((F |` x) Fn x /\ Fun `'(F |` x) /\ ran ( F |` x) = A))
14		df-3an 775	. . . . . . . 8 |- (((F |` x) Fn x /\ Fun `'(F |` x) /\ ran ( F |` x) = A) <-> (((F |` x) Fn x /\ Fun `'(F |` x)) /\ ran ( F |` x) = A))
15		an23 484	. . . . . . . 8 |- ((((F |` x) Fn x /\ Fun `'(F |` x)) /\ ran ( F |` x) = A) <-> (((F |` x) Fn x /\ ran ( F |` x) = A) /\ Fun `'(F |` x)))
16	13, 14, 15	3bitr 177	. . . . . . 7 |- ((F |` x):x-1-1-onto->A <-> (((F |` x) Fn x /\ ran ( F |` x) = A) /\ Fun `'(F |` x)))
17	12, 16	sylibr 200	. . . . . 6 |- (((x e. On /\ (F"x) = A) /\ Fun `'(F |` x)) -> (F |` x):x-1-1-onto->A)
18	17	exp31 376	. . . . 5 |- (x e. On -> ((F"x) = A -> (Fun `'(F |` x) -> (F |` x):x-1-1-onto->A)))
19	18	imp3a 361	. . . 4 |- (x e. On -> (((F"x) = A /\ Fun `'(F |` x)) -> (F |` x):x-1-1-onto->A))
20		3simpc 785	. . . 4 |- ((A.y e. x (A \ (F"y)) =/= (/) /\ (F"x) = A /\ Fun `'(F |` x)) -> ((F"x) = A /\ Fun `'(F |` x)))
21	19, 20	syl5 21	. . 3 |- (x e. On -> ((A.y e. x (A \ (F"y)) =/= (/) /\ (F"x) = A /\ Fun `'(F |` x)) -> (F |` x):x-1-1-onto->A))
22	21	r19.22i 1724	. 2 |- (E.x e. On (A.y e. x (A \ (F"y)) =/= (/) /\ (F"x) = A /\ Fun `'(F |` x)) -> E.x e. On (F |` x):x-1-1-onto->A)
23	3, 22	syl 10	1 |- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (F |` x):x-1-1-onto->A)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  E.wrex 1638  Vcvv 1802   \ cdif 2034   (_ wss 2037  (/)c0 2270  Oncon0 2938  `'ccnv 3159  ran crn 3161   |` cres 3162  "cima 3163  Fun wfun 3166   Fn wfn 3167  -1-1-onto->wf1o 3171  ` cfv 3172

This theorem is referenced by:  numthlem 4755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188

Copyright terms: Public domain