Theorem wfrlem16 13972

Description: Lemma for well-founded recursion. If z is R minimal in (A \ dom F), then C is acceptable and thus a subset of F, but dom C is bigger than dom F. Thus, z cannot be minimal, so (A \ dom F) must be empty, and (due to wfrlem7 13963), dom F = A.

Hypotheses

Ref	Expression
wfrlem13.1	|- R We A
wfrlem13.2	|- A.x e. A Pred(R, A, x) e. _V
wfrlem13.3	|- B = {f | E.x(f Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred(R, A, y))))}
wfrlem13.4	|- F = U.B
wfrlem13.5	|- C = (F u. {<.z, (G` (F |` Pred(R, A, z)))>.})

Assertion

Ref	Expression
wfrlem16	|- dom F = A

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

Proof of Theorem wfrlem16

Step	Hyp	Ref	Expression
1		wfrlem13.3	. . 3 |- B = {f | E.x(f Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred(R, A, y))))}
2		wfrlem13.4	. . 3 |- F = U.B
3	1, 2	wfrlem7 13963	. 2 |- dom F C_ A
4		eldifn 2731	. . . . . 6 |- (z e. (A \ dom F) -> -. z e. dom F)
5		ssun2 2768	. . . . . . . . 9 |- {z} C_ (dom F u. {z})
6		visset 2295	. . . . . . . . . 10 |- z e. _V
7	6	snid 3069	. . . . . . . . 9 |- z e. {z}
8	5, 7	sselii 2618	. . . . . . . 8 |- z e. (dom F u. {z})
9		wfrlem13.5	. . . . . . . . . 10 |- C = (F u. {<.z, (G` (F |` Pred(R, A, z)))>.})
10	9	dmeqi 4158	. . . . . . . . 9 |- dom C = dom ( F u. {<.z, (G` (F |` Pred(R, A, z)))>.})
11		dmun 4163	. . . . . . . . 9 |- dom ( F u. {<.z, (G` (F |` Pred(R, A, z)))>.}) = (dom F u. dom {<.z, (G` (F |` Pred(R, A, z)))>.})
12		dmsnop 4367	. . . . . . . . . 10 |- dom {<.z, (G` (F |` Pred(R, A, z)))>.} = {z}
13	12	uneq2i 2752	. . . . . . . . 9 |- (dom F u. dom {<.z, (G` (F |` Pred(R, A, z)))>.}) = (dom F u. {z})
14	10, 11, 13	3eqtri 1912	. . . . . . . 8 |- dom C = (dom F u. {z})
15	8, 14	eleqtrri 1970	. . . . . . 7 |- z e. dom C
16		wfrlem13.1	. . . . . . . . . . . 12 |- R We A
17		wfrlem13.2	. . . . . . . . . . . 12 |- A.x e. A Pred(R, A, x) e. _V
18	16, 17, 1, 2, 9	wfrlem15 13971	. . . . . . . . . . 11 |- ((z e. (A \ dom F) /\ Pred(R, (A \ dom F), z) = (/)) -> C e. B)
19		elssuni 3206	. . . . . . . . . . 11 |- (C e. B -> C C_ U.B)
20	18, 19	syl 12	. . . . . . . . . 10 |- ((z e. (A \ dom F) /\ Pred(R, (A \ dom F), z) = (/)) -> C C_ U.B)
21	20, 2	syl6ssr 2664	. . . . . . . . 9 |- ((z e. (A \ dom F) /\ Pred(R, (A \ dom F), z) = (/)) -> C C_ F)
22		dmss 4156	. . . . . . . . 9 |- (C C_ F -> dom C C_ dom F)
23	21, 22	syl 12	. . . . . . . 8 |- ((z e. (A \ dom F) /\ Pred(R, (A \ dom F), z) = (/)) -> dom C C_ dom F)
24	23	sseld 2619	. . . . . . 7 |- ((z e. (A \ dom F) /\ Pred(R, (A \ dom F), z) = (/)) -> (z e. dom C -> z e. dom F))
25	15, 24	mpi 55	. . . . . 6 |- ((z e. (A \ dom F) /\ Pred(R, (A \ dom F), z) = (/)) -> z e. dom F)
26	4, 25	mtand 520	. . . . 5 |- (z e. (A \ dom F) -> -. Pred(R, (A \ dom F), z) = (/))
27	26	nrex 2192	. . . 4 |- -. E.z e. (A \ dom F)Pred(R, (A \ dom F), z) = (/)
28		df-ne 2019	. . . . 5 |- ((A \ dom F) =/= (/) <-> -. (A \ dom F) = (/))
29		difss 2735	. . . . . 6 |- (A \ dom F) C_ A
30	16, 17	tz6.26i 13914	. . . . . 6 |- (((A \ dom F) C_ A /\ (A \ dom F) =/= (/)) -> E.z e. (A \ dom F)Pred(R, (A \ dom F), z) = (/))
31	29, 30	mpan 759	. . . . 5 |- ((A \ dom F) =/= (/) -> E.z e. (A \ dom F)Pred(R, (A \ dom F), z) = (/))
32	28, 31	sylbir 218	. . . 4 |- (-. (A \ dom F) = (/) -> E.z e. (A \ dom F)Pred(R, (A \ dom F), z) = (/))
33	27, 32	mt3 127	. . 3 |- (A \ dom F) = (/)
34		ssdif0 2934	. . 3 |- (A C_ dom F <-> (A \ dom F) = (/))
35	33, 34	mpbir 207	. 2 |- A C_ dom F
36	3, 35	eqssi 2632	1 |- dom F = A

Syntax hints:  -. wn 2   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  (/)c0 2875  {csn 3044  <.cop 3046  U.cuni 3177   We wwe 3624  dom cdm 3986   |` cres 3988   Fn wfn 3993  ` cfv 3998  Predcpred 13879

This theorem is referenced by:  wfr1 13973  wfr2 13974

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-pred 13880

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