Theorem tfrlem10 4221

Description: Lemma for transfinite recursion. We define class C by extending F with one ordered pair. We will assume, falsely, that domain of F is a member of, and thus not equal to, On. Using this assumption we will prove facts about C that will lead to a contradiction in tfrlem13 4224, thus showing the domain of F does in fact equal On. Here we show (under the false assumption) that C is a function extending the domain of F by one. (The proof was shortened by Alan Sare, 20-Feb-2008.)

Hypotheses

Ref	Expression
tfrlem.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfrlem.2	|- F = U.A
tfrlem.3	|- C = (F u. {<.dom F, (G` (F |` dom F))>.})

Assertion

Ref	Expression
tfrlem10	|- (dom F e. On -> C Fn suc dom F)

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f   x,C,y,f

Proof of Theorem tfrlem10

Step	Hyp	Ref	Expression
1		funun 3659	. . . . 5 |- (((Fun F /\ Fun {<.dom F, (G` (F |` dom F))>.}) /\ (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (/)) -> Fun (F u. {<.dom F, (G` (F |` dom F))>.}))
2		opeq1 2552	. . . . . . . 8 |- (w = dom F -> <.w, (G` (F |` dom F))>. = <.dom F, (G` (F |` dom F))>.)
3		sneq 2475	. . . . . . . 8 |- (<.w, (G` (F |` dom F))>. = <.dom F, (G` (F |` dom F))>. -> {<.w, (G` (F |` dom F))>.} = {<.dom F, (G` (F |` dom F))>.})
4		funeq 3640	. . . . . . . 8 |- ({<.w, (G` (F |` dom F))>.} = {<.dom F, (G` (F |` dom F))>.} -> (Fun {<.w, (G` (F |` dom F))>.} <-> Fun {<.dom F, (G` (F |` dom F))>.}))
5	2, 3, 4	3syl 20	. . . . . . 7 |- (w = dom F -> (Fun {<.w, (G` (F |` dom F))>.} <-> Fun {<.dom F, (G` (F |` dom F))>.}))
6		visset 1859	. . . . . . . 8 |- w e. V
7		fvex 3843	. . . . . . . 8 |- (G` (F |` dom F)) e. V
8	6, 7	funsn 3648	. . . . . . 7 |- Fun {<.w, (G` (F |` dom F))>.}
9	5, 8	vtoclg 1893	. . . . . 6 |- (dom F e. On -> Fun {<.dom F, (G` (F |` dom F))>.})
10		tfrlem.1	. . . . . . 7 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
11		tfrlem.2	. . . . . . 7 |- F = U.A
12	10, 11	tfrlem7 4218	. . . . . 6 |- Fun F
13	9, 12	jctil 290	. . . . 5 |- (dom F e. On -> (Fun F /\ Fun {<.dom F, (G` (F |` dom F))>.}))
14		dmsnop 3577	. . . . . . . 8 |- dom {<.dom F, (G` (F |` dom F))>.} = {dom F}
15	14	ineq2i 2266	. . . . . . 7 |- (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (dom F i^i {dom F})
16	10, 11	tfrlem8 4219	. . . . . . . 8 |- Ord dom F
17		orddisj 3013	. . . . . . . 8 |- (Ord dom F -> (dom F i^i {dom F}) = (/))
18	16, 17	ax-mp 7	. . . . . . 7 |- (dom F i^i {dom F}) = (/)
19	15, 18	eqtri 1538	. . . . . 6 |- (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (/)
20	19	a1i 8	. . . . 5 |- (dom F e. On -> (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (/))
21	1, 13, 20	sylanc 473	. . . 4 |- (dom F e. On -> Fun (F u. {<.dom F, (G` (F |` dom F))>.}))
22	14	uneq2i 2233	. . . . 5 |- (dom F u. dom {<.dom F, (G` (F |` dom F))>.}) = (dom F u. {dom F})
23		dmun 3408	. . . . 5 |- dom ( F u. {<.dom F, (G` (F |` dom F))>.}) = (dom F u. dom {<.dom F, (G` (F |` dom F))>.})
24		df-suc 2981	. . . . 5 |- suc dom F = (dom F u. {dom F})
25	22, 23, 24	3eqtr4i 1548	. . . 4 |- dom ( F u. {<.dom F, (G` (F |` dom F))>.}) = suc dom F
26	21, 25	jctir 291	. . 3 |- (dom F e. On -> (Fun (F u. {<.dom F, (G` (F |` dom F))>.}) /\ dom ( F u. {<.dom F, (G` (F |` dom F))>.}) = suc dom F))
27		df-fn 3274	. . 3 |- ((F u. {<.dom F, (G` (F |` dom F))>.}) Fn suc dom F <-> (Fun (F u. {<.dom F, (G` (F |` dom F))>.}) /\ dom ( F u. {<.dom F, (G` (F |` dom F))>.}) = suc dom F))
28	26, 27	sylibr 198	. 2 |- (dom F e. On -> (F u. {<.dom F, (G` (F |` dom F))>.}) Fn suc dom F)
29		tfrlem.3	. . 3 |- C = (F u. {<.dom F, (G` (F |` dom F))>.})
30		fneq1 3688	. . 3 |- (C = (F u. {<.dom F, (G` (F |` dom F))>.}) -> (C Fn suc dom F <-> (F u. {<.dom F, (G` (F |` dom F))>.}) Fn suc dom F))
31	29, 30	ax-mp 7	. 2 |- (C Fn suc dom F <-> (F u. {<.dom F, (G` (F |` dom F))>.}) Fn suc dom F)
32	28, 31	sylibr 198	1 |- (dom F e. On -> C Fn suc dom F)

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  {cab 1505  A.wral 1691  E.wrex 1692   u. cun 2097   i^i cin 2098  (/)c0 2332  {csn 2467  <.cop 2469  U.cuni 2569  Ord word 2974  Oncon0 2975  suc csuc 2977  dom cdm 3251   |` cres 3253  Fun wfun 3257   Fn wfn 3258  ` cfv 3263

This theorem is referenced by:  tfrlem11 4222  tfrlem12 4223  tfrlem13 4224

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279

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