Theorem tfrlem10 3911

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 3914, 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 3546	. . . . 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 2483	. . . . . . . 8 |- (w = dom F -> <.w, (G` (F |` dom F))>. = <.dom F, (G` (F |` dom F))>.)
3		sneq 2413	. . . . . . . 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 3527	. . . . . . . 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 1809	. . . . . . . 8 |- w e. V
7		fvex 3723	. . . . . . . 8 |- (G` (F |` dom F)) e. V
8	6, 7	funsn 3535	. . . . . . 7 |- Fun {<.w, (G` (F |` dom F))>.}
9	5, 8	vtoclg 1843	. . . . . 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 3908	. . . . . 6 |- Fun F
13	9, 12	jctil 292	. . . . 5 |- (dom F e. On -> (Fun F /\ Fun {<.dom F, (G` (F |` dom F))>.}))
14		dmsnop 3323	. . . . . . . 8 |- dom {<.dom F, (G` (F |` dom F))>.} = {dom F}
15	14	ineq2i 2210	. . . . . . 7 |- (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (dom F i^i {dom F})
16	10, 11	tfrlem8 3909	. . . . . . . 8 |- Ord dom F
17		orddisj 2980	. . . . . . . 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	eqtr 1492	. . . . . 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 471	. . . 4 |- (dom F e. On -> Fun (F u. {<.dom F, (G` (F |` dom F))>.}))
22	14	uneq2i 2177	. . . . 5 |- (dom F u. dom {<.dom F, (G` (F |` dom F))>.}) = (dom F u. {dom F})
23		dmun 3312	. . . . 5 |- dom ( F u. {<.dom F, (G` (F |` dom F))>.}) = (dom F u. dom {<.dom F, (G` (F |` dom F))>.})
24		df-suc 2949	. . . . 5 |- suc dom F = (dom F u. {dom F})
25	22, 23, 24	3eqtr4 1502	. . . 4 |- dom ( F u. {<.dom F, (G` (F |` dom F))>.}) = suc dom F
26	21, 25	jctir 293	. . 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 3188	. . 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 200	. 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 3574	. . 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 200	1 |- (dom F e. On -> C Fn suc dom F)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  A.wral 1642  E.wrex 1643   u. cun 2041   i^i cin 2042  (/)c0 2276  {csn 2405  <.cop 2407  U.cuni 2498  Ord word 2942  Oncon0 2943  suc csuc 2945  dom cdm 3165   |` cres 3167  Fun wfun 3171   Fn wfn 3172  ` cfv 3177

This theorem is referenced by:  tfrlem11 3912  tfrlem12 3913  tfrlem13 3914

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-sep 2698  ax-nul 2705  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-rab 1649  df-v 1808  df-sbc 1938  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-iun 2563  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-suc 2949  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-fv 3193

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