Theorem tz7.48-1 3956

Description: Proposition 7.48(1) of [TakeutiZaring] p. 51.

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 (_ A)

Distinct variable groups:   x,F   x,A

Proof of Theorem tz7.48-1

Step	Hyp	Ref	Expression
1		hbra1 1687	. . . 4 |- (A.x e. On (F` x) e. (A \ (F"x)) -> A.xA.x e. On (F` x) e. (A \ (F"x)))
2		ax-17 971	. . . 4 |- (y e. A -> A.x y e. A)
3		ra4 1694	. . . . 5 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (x e. On -> (F` x) e. (A \ (F"x))))
4		eleq1 1534	. . . . . . . 8 |- ((F` x) = y -> ((F` x) e. A <-> y e. A))
5		eldifi 2162	. . . . . . . 8 |- ((F` x) e. (A \ (F"x)) -> (F` x) e. A)
6	4, 5	syl5cbi 209	. . . . . . 7 |- ((F` x) e. (A \ (F"x)) -> ((F` x) = y -> y e. A))
7	6	imim2i 17	. . . . . 6 |- ((x e. On -> (F` x) e. (A \ (F"x))) -> (x e. On -> ((F` x) = y -> y e. A)))
8	7	imp3a 361	. . . . 5 |- ((x e. On -> (F` x) e. (A \ (F"x))) -> ((x e. On /\ (F` x) = y) -> y e. A))
9	3, 8	syl 10	. . . 4 |- (A.x e. On (F` x) e. (A \ (F"x)) -> ((x e. On /\ (F` x) = y) -> y e. A))
10	1, 2, 9	19.23ad 1066	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (E.x(x e. On /\ (F` x) = y) -> y e. A))
11		visset 1813	. . . . 5 |- y e. V
12	11	elrn2 3349	. . . 4 |- (y e. ran F <-> E.x<.x, y>. e. F)
13		visset 1813	. . . . . . . . 9 |- x e. V
14	13	opeldm 3314	. . . . . . . 8 |- (<.x, y>. e. F -> x e. dom F)
15		tz7.48.1	. . . . . . . . 9 |- F Fn On
16		fndm 3587	. . . . . . . . 9 |- (F Fn On -> dom F = On)
17	15, 16	ax-mp 7	. . . . . . . 8 |- dom F = On
18	14, 17	syl6eleq 1558	. . . . . . 7 |- (<.x, y>. e. F -> x e. On)
19	18	ancri 297	. . . . . 6 |- (<.x, y>. e. F -> (x e. On /\ <.x, y>. e. F))
20	11	fnopfvb 3754	. . . . . . . 8 |- ((F Fn On /\ x e. On) -> ((F` x) = y <-> <.x, y>. e. F))
21	15, 20	mpan 695	. . . . . . 7 |- (x e. On -> ((F` x) = y <-> <.x, y>. e. F))
22	21	pm5.32i 645	. . . . . 6 |- ((x e. On /\ (F` x) = y) <-> (x e. On /\ <.x, y>. e. F))
23	19, 22	sylibr 200	. . . . 5 |- (<.x, y>. e. F -> (x e. On /\ (F` x) = y))
24	23	19.22i 1040	. . . 4 |- (E.x<.x, y>. e. F -> E.x(x e. On /\ (F` x) = y))
25	12, 24	sylbi 199	. . 3 |- (y e. ran F -> E.x(x e. On /\ (F` x) = y))
26	10, 25	syl5 21	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (y e. ran F -> y e. A))
27	26	ssrdv 2070	1 |- (A.x e. On (F` x) e. (A \ (F"x)) -> ran F (_ A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  A.wral 1645   \ cdif 2044   (_ wss 2047  <.cop 2411  Oncon0 2948  dom cdm 3170  ran crn 3171  "cima 3173   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  tz7.48-3 3958

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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