Theorem inf3lem6 4618

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 4620 for detailed description.

Hypotheses

Ref	Expression
inf3lem.1	|- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
inf3lem.2	|- F = (rec(G, (/)) |` om)
inf3lem.3	|- A e. V
inf3lem.4	|- B e. V

Assertion

Ref	Expression
inf3lem6	|- ((x =/= (/) /\ x (_ U.x) -> F:om-1-1->P~x)

Distinct variable group:   x,y,z,w

Proof of Theorem inf3lem6

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . . . . . . . . . 14 |- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
2		inf3lem.2	. . . . . . . . . . . . . 14 |- F = (rec(G, (/)) |` om)
3		visset 1813	. . . . . . . . . . . . . 14 |- u e. V
4		visset 1813	. . . . . . . . . . . . . 14 |- v e. V
5	1, 2, 3, 4	inf3lem5 4617	. . . . . . . . . . . . 13 |- ((x =/= (/) /\ x (_ U.x) -> ((u e. om /\ v e. u) -> (F` v) (. (F` u)))
6		dfpss2 2133	. . . . . . . . . . . . . 14 |- ((F` v) (. (F` u) <-> ((F` v) (_ (F` u) /\ -. (F` v) = (F` u)))
7	6	pm3.27bi 326	. . . . . . . . . . . . 13 |- ((F` v) (. (F` u) -> -. (F` v) = (F` u))
8	5, 7	syl6 22	. . . . . . . . . . . 12 |- ((x =/= (/) /\ x (_ U.x) -> ((u e. om /\ v e. u) -> -. (F` v) = (F` u)))
9	8	exp3a 375	. . . . . . . . . . 11 |- ((x =/= (/) /\ x (_ U.x) -> (u e. om -> (v e. u -> -. (F` v) = (F` u))))
10	9	imp 350	. . . . . . . . . 10 |- (((x =/= (/) /\ x (_ U.x) /\ u e. om) -> (v e. u -> -. (F` v) = (F` u)))
11	10	adantrl 394	. . . . . . . . 9 |- (((x =/= (/) /\ x (_ U.x) /\ (v e. om /\ u e. om)) -> (v e. u -> -. (F` v) = (F` u)))
12	1, 2, 4, 3	inf3lem5 4617	. . . . . . . . . . . . 13 |- ((x =/= (/) /\ x (_ U.x) -> ((v e. om /\ u e. v) -> (F` u) (. (F` v)))
13		dfpss2 2133	. . . . . . . . . . . . . . 15 |- ((F` u) (. (F` v) <-> ((F` u) (_ (F` v) /\ -. (F` u) = (F` v)))
14	13	pm3.27bi 326	. . . . . . . . . . . . . 14 |- ((F` u) (. (F` v) -> -. (F` u) = (F` v))
15		eqcom 1477	. . . . . . . . . . . . . . 15 |- ((F` u) = (F` v) <-> (F` v) = (F` u))
16	15	negbii 187	. . . . . . . . . . . . . 14 |- (-. (F` u) = (F` v) <-> -. (F` v) = (F` u))
17	14, 16	sylib 198	. . . . . . . . . . . . 13 |- ((F` u) (. (F` v) -> -. (F` v) = (F` u))
18	12, 17	syl6 22	. . . . . . . . . . . 12 |- ((x =/= (/) /\ x (_ U.x) -> ((v e. om /\ u e. v) -> -. (F` v) = (F` u)))
19	18	exp3a 375	. . . . . . . . . . 11 |- ((x =/= (/) /\ x (_ U.x) -> (v e. om -> (u e. v -> -. (F` v) = (F` u))))
20	19	imp 350	. . . . . . . . . 10 |- (((x =/= (/) /\ x (_ U.x) /\ v e. om) -> (u e. v -> -. (F` v) = (F` u)))
21	20	adantrr 395	. . . . . . . . 9 |- (((x =/= (/) /\ x (_ U.x) /\ (v e. om /\ u e. om)) -> (u e. v -> -. (F` v) = (F` u)))
22	11, 21	jaod 424	. . . . . . . 8 |- (((x =/= (/) /\ x (_ U.x) /\ (v e. om /\ u e. om)) -> ((v e. u \/ u e. v) -> -. (F` v) = (F` u)))
23	22	con2d 91	. . . . . . 7 |- (((x =/= (/) /\ x (_ U.x) /\ (v e. om /\ u e. om)) -> ((F` v) = (F` u) -> -. (v e. u \/ u e. v)))
24		ordtri3 2983	. . . . . . . . 9 |- ((Ord v /\ Ord u) -> (v = u <-> -. (v e. u \/ u e. v)))
25		nnord 3140	. . . . . . . . 9 |- (v e. om -> Ord v)
26		nnord 3140	. . . . . . . . 9 |- (u e. om -> Ord u)
27	24, 25, 26	syl2an 454	. . . . . . . 8 |- ((v e. om /\ u e. om) -> (v = u <-> -. (v e. u \/ u e. v)))
28	27	adantl 388	. . . . . . 7 |- (((x =/= (/) /\ x (_ U.x) /\ (v e. om /\ u e. om)) -> (v = u <-> -. (v e. u \/ u e. v)))
29	23, 28	sylibrd 204	. . . . . 6 |- (((x =/= (/) /\ x (_ U.x) /\ (v e. om /\ u e. om)) -> ((F` v) = (F` u) -> v = u))
30	29	ex 373	. . . . 5 |- ((x =/= (/) /\ x (_ U.x) -> ((v e. om /\ u e. om) -> ((F` v) = (F` u) -> v = u)))
31	30	19.21aivv 1287	. . . 4 |- ((x =/= (/) /\ x (_ U.x) -> A.vA.u((v e. om /\ u e. om) -> ((F` v) = (F` u) -> v = u)))
32		r2al 1676	. . . 4 |- (A.v e. om A.u e. om ((F` v) = (F` u) -> v = u) <-> A.vA.u((v e. om /\ u e. om) -> ((F` v) = (F` u) -> v = u)))
33	31, 32	sylibr 200	. . 3 |- ((x =/= (/) /\ x (_ U.x) -> A.v e. om A.u e. om ((F` v) = (F` u) -> v = u))
34		frfnom 3951	. . . . . . 7 |- (rec(G, (/)) |` om) Fn om
35		fneq1 3582	. . . . . . 7 |- (F = (rec(G, (/)) |` om) -> (F Fn om <-> (rec(G, (/)) |` om) Fn om))
36	34, 35	mpbiri 194	. . . . . 6 |- (F = (rec(G, (/)) |` om) -> F Fn om)
37	2, 36	ax-mp 7	. . . . 5 |- F Fn om
38		fvelrnb 3760	. . . . . . . 8 |- (F Fn om -> (u e. ran F <-> E.v e. om (F` v) = u))
39		eleq1 1534	. . . . . . . . . 10 |- ((F` v) = u -> ((F` v) e. P~x <-> u e. P~x))
40		inf3lem.4	. . . . . . . . . . . 12 |- B e. V
41	1, 2, 4, 40	inf3lemd 4612	. . . . . . . . . . 11 |- (v e. om -> (F` v) (_ x)
42		fvex 3732	. . . . . . . . . . . 12 |- (F` v) e. V
43	42	elpw 2404	. . . . . . . . . . 11 |- ((F` v) e. P~x <-> (F` v) (_ x)
44	41, 43	sylibr 200	. . . . . . . . . 10 |- (v e. om -> (F` v) e. P~x)
45	39, 44	syl5cbi 209	. . . . . . . . 9 |- (v e. om -> ((F` v) = u -> u e. P~x))
46	45	r19.23aiv 1743	. . . . . . . 8 |- (E.v e. om (F` v) = u -> u e. P~x)
47	38, 46	syl6bi 214	. . . . . . 7 |- (F Fn om -> (u e. ran F -> u e. P~x))
48	47	ssrdv 2070	. . . . . 6 |- (F Fn om -> ran F (_ P~x)
49	48	ancli 296	. . . . 5 |- (F Fn om -> (F Fn om /\ ran F (_ P~x))
50	37, 49	ax-mp 7	. . . 4 |- (F Fn om /\ ran F (_ P~x)
51		df-f 3194	. . . 4 |- (F:om-->P~x <-> (F Fn om /\ ran F (_ P~x))
52	50, 51	mpbir 190	. . 3 |- F:om-->P~x
53	33, 52	jctil 292	. 2 |- ((x =/= (/) /\ x (_ U.x) -> (F:om-->P~x /\ A.v e. om A.u e. om ((F` v) = (F` u) -> v = u)))
54		f1fv 3874	. 2 |- (F:om-1-1->P~x <-> (F:om-->P~x /\ A.v e. om A.u e. om ((F` v) = (F` u) -> v = u)))
55	53, 54	sylibr 200	1 |- ((x =/= (/) /\ x (_ U.x) -> F:om-1-1->P~x)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646  {crab 1648  Vcvv 1811   i^i cin 2046   (_ wss 2047   (. wpss 2048  (/)c0 2280  P~cpw 2401  U.cuni 2503  {copab 2666  Ord word 2947  omcom 3131  ran crn 3171   |` cres 3172   Fn wfn 3177  -->wf 3178  -1-1->wf1 3179  ` cfv 3182  reccrdg 3931

This theorem is referenced by:  inf3lem7 4619  dominf 4904

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  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-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  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-f 3194  df-f1 3195  df-fv 3198  df-rdg 3932

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