Theorem hgralem 10714

Description: Lemma for various hypergraph theorems.

Hypotheses

Ref	Expression
hgralem.1	|- A = (1st` H)
hgralem.2	|- B = (2nd` H)

Assertion

Ref	Expression
hgralem	|- (H e. HypGrph -> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)})))

Proof of Theorem hgralem

Step	Hyp	Ref	Expression
1		df-hgra 10710	. . 3 |- HypGrph = {<.x, y>. | ((x i^i y) = (/) /\ y (_ (P~x \ {(/)}))}
2	1	eleq2i 1537	. 2 |- (H e. HypGrph <-> H e. {<.x, y>. | ((x i^i y) = (/) /\ y (_ (P~x \ {(/)}))})
3		hgralem.1	. . . . . . 7 |- A = (1st` H)
4	3	eqeq2i 1484	. . . . . 6 |- (x = A <-> x = (1st` H))
5		ineq1 2208	. . . . . 6 |- (x = A -> (x i^i y) = (A i^i y))
6	4, 5	sylbir 201	. . . . 5 |- (x = (1st` H) -> (x i^i y) = (A i^i y))
7	6	eqeq1d 1482	. . . 4 |- (x = (1st` H) -> ((x i^i y) = (/) <-> (A i^i y) = (/)))
8		pweq 2401	. . . . . . 7 |- (x = A -> P~x = P~A)
9	8	difeq1d 2156	. . . . . 6 |- (x = A -> (P~x \ {(/)}) = (P~A \ {(/)}))
10	9	sseq2d 2087	. . . . 5 |- (x = A -> (y (_ (P~x \ {(/)}) <-> y (_ (P~A \ {(/)})))
11	4, 10	sylbir 201	. . . 4 |- (x = (1st` H) -> (y (_ (P~x \ {(/)}) <-> y (_ (P~A \ {(/)})))
12	7, 11	anbi12d 627	. . 3 |- (x = (1st` H) -> (((x i^i y) = (/) /\ y (_ (P~x \ {(/)})) <-> ((A i^i y) = (/) /\ y (_ (P~A \ {(/)}))))
13		hgralem.2	. . . . . . 7 |- B = (2nd` H)
14	13	eqeq2i 1484	. . . . . 6 |- (y = B <-> y = (2nd` H))
15		ineq2 2209	. . . . . 6 |- (y = B -> (A i^i y) = (A i^i B))
16	14, 15	sylbir 201	. . . . 5 |- (y = (2nd` H) -> (A i^i y) = (A i^i B))
17	16	eqeq1d 1482	. . . 4 |- (y = (2nd` H) -> ((A i^i y) = (/) <-> (A i^i B) = (/)))
18		sseq1 2080	. . . . 5 |- (y = B -> (y (_ (P~A \ {(/)}) <-> B (_ (P~A \ {(/)})))
19	14, 18	sylbir 201	. . . 4 |- (y = (2nd` H) -> (y (_ (P~A \ {(/)}) <-> B (_ (P~A \ {(/)})))
20	17, 19	anbi12d 627	. . 3 |- (y = (2nd` H) -> (((A i^i y) = (/) /\ y (_ (P~A \ {(/)})) <-> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)}))))
21	12, 20	elopabi 4114	. 2 |- (H e. {<.x, y>. | ((x i^i y) = (/) /\ y (_ (P~x \ {(/)}))} -> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)})))
22	2, 21	sylbi 199	1 |- (H e. HypGrph -> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)})))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957   \ cdif 2042   i^i cin 2044   (_ wss 2045  (/)c0 2278  P~cpw 2399  {csn 2407  {copab 2663  ` cfv 3179  1stc1st 4074  2ndc2nd 4075  HypGrphchgra 10709

This theorem is referenced by:  hgradj 10715  hgrablkconn 10716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fv 3195  df-1st 4076  df-2nd 4077  df-hgra 10710

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