Theorem tz7.49c 3951

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 3950	. 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 2987	. . . . . . . . . 10 |- (x e. On -> x (_ On)
5		fnssres 3592	. . . . . . . . . . 11 |- ((F Fn On /\ x (_ On) -> (F |` x) Fn x)
6	1, 5	mpan 694	. . . . . . . . . 10 |- (x (_ On -> (F |` x) Fn x)
7	4, 6	syl 10	. . . . . . . . 9 |- (x e. On -> (F |` x) Fn x)
8		df-ima 3186	. . . . . . . . . . 11 |- (F"x) = ran ( F |` x)
9	8	eqeq1i 1479	. . . . . . . . . 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 3684	. . . . . . . 8 |- ((F |` x):x-1-1-onto->A <-> ((F |` x) Fn x /\ Fun `'(F |` x) /\ ran ( F |` x) = A))
14		df-3an 776	. . . . . . . 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 485	. . . . . . . 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 786	. . . 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 1729	. 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 774   = wceq 954   e. wcel 956   =/= wne 1582  A.wral 1642  E.wrex 1643  Vcvv 1807   \ cdif 2040   (_ wss 2043  (/)c0 2276  Oncon0 2943  `'ccnv 3164  ran crn 3166   |` cres 3167  "cima 3168  Fun wfun 3171   Fn wfn 3172  -1-1-onto->wf1o 3176  ` cfv 3177

This theorem is referenced by:  numthlem 4763

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193

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