Theorem ndmoprcl 3058

Description: The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases.

Hypotheses

Ref	Expression
ndmoprcl.1	⊢ dom F = (S × S)
ndmoprcl.2	⊢ ((A ∈ S ∧ x ∈ S) → (AFx) ∈ S)
ndmoprcl.3	⊢ ∅ ∈ S

Assertion

Ref	Expression
ndmoprcl	⊢ (AFB) ∈ S

Distinct variable group(s):   x,A   x,B   x,F   x,S

Proof of Theorem ndmoprcl

Step	Hyp	Ref	Expression
1		oprprc2 3020	. . . . . 6 ⊢ (¬ B ∈ V → (AFB) = (AFA))
2	1	eleq1d 1155	. . . . 5 ⊢ (¬ B ∈ V → ((AFB) ∈ S ↔ (AFA) ∈ S))
3		ndmoprcl.3	. . . . . . . 8 ⊢ ∅ ∈ S
4		ndmoprcl.1	. . . . . . . . . 10 ⊢ dom F = (S × S)
5	4	ndmoprg 3057	. . . . . . . . 9 ⊢ ((A ∈ V ∧ ¬ (A ∈ S ∧ A ∈ S)) → (AFA) = ∅)
6	5	eleq1d 1155	. . . . . . . 8 ⊢ ((A ∈ V ∧ ¬ (A ∈ S ∧ A ∈ S)) → ((AFA) ∈ S ↔ ∅ ∈ S))
7	3, 6	mpbiri 169	. . . . . . 7 ⊢ ((A ∈ V ∧ ¬ (A ∈ S ∧ A ∈ S)) → (AFA) ∈ S)
8	7	exp 291	. . . . . 6 ⊢ (A ∈ V → (¬ (A ∈ S ∧ A ∈ S) → (AFA) ∈ S))
9		opreq2 3007	. . . . . . . . . 10 ⊢ (x = A → (AFx) = (AFA))
10	9	eleq1d 1155	. . . . . . . . 9 ⊢ (x = A → ((AFx) ∈ S ↔ (AFA) ∈ S))
11	10	imbi2d 464	. . . . . . . 8 ⊢ (x = A → ((A ∈ S → (AFx) ∈ S) ↔ (A ∈ S → (AFA) ∈ S)))
12		ndmoprcl.2	. . . . . . . . . 10 ⊢ ((A ∈ S ∧ x ∈ S) → (AFx) ∈ S)
13	12	exp 291	. . . . . . . . 9 ⊢ (A ∈ S → (x ∈ S → (AFx) ∈ S))
14	13	com12 13	. . . . . . . 8 ⊢ (x ∈ S → (A ∈ S → (AFx) ∈ S))
15	11, 14	vtoclga 1387	. . . . . . 7 ⊢ (A ∈ S → (A ∈ S → (AFA) ∈ S))
16	15	imp 277	. . . . . 6 ⊢ ((A ∈ S ∧ A ∈ S) → (AFA) ∈ S)
17	8, 16	pm2.61d2 111	. . . . 5 ⊢ (A ∈ V → (AFA) ∈ S)
18	2, 17	syl5bir 184	. . . 4 ⊢ (¬ B ∈ V → (A ∈ V → (AFB) ∈ S))
19	18	com12 13	. . 3 ⊢ (A ∈ V → (¬ B ∈ V → (AFB) ∈ S))
20	4	ndmoprg 3057	. . . . . . 7 ⊢ ((B ∈ V ∧ ¬ (A ∈ S ∧ B ∈ S)) → (AFB) = ∅)
21	20	eleq1d 1155	. . . . . 6 ⊢ ((B ∈ V ∧ ¬ (A ∈ S ∧ B ∈ S)) → ((AFB) ∈ S ↔ ∅ ∈ S))
22	3, 21	mpbiri 169	. . . . 5 ⊢ ((B ∈ V ∧ ¬ (A ∈ S ∧ B ∈ S)) → (AFB) ∈ S)
23	22	exp 291	. . . 4 ⊢ (B ∈ V → (¬ (A ∈ S ∧ B ∈ S) → (AFB) ∈ S))
24		opreq2 3007	. . . . . . . . 9 ⊢ (x = B → (AFx) = (AFB))
25	24	eleq1d 1155	. . . . . . . 8 ⊢ (x = B → ((AFx) ∈ S ↔ (AFB) ∈ S))
26	25	imbi2d 464	. . . . . . 7 ⊢ (x = B → ((A ∈ S → (AFx) ∈ S) ↔ (A ∈ S → (AFB) ∈ S)))
27	26, 14	vtoclga 1387	. . . . . 6 ⊢ (B ∈ S → (A ∈ S → (AFB) ∈ S))
28	27	com12 13	. . . . 5 ⊢ (A ∈ S → (B ∈ S → (AFB) ∈ S))
29	28	imp 277	. . . 4 ⊢ ((A ∈ S ∧ B ∈ S) → (AFB) ∈ S)
30	23, 29	pm2.61d2 111	. . 3 ⊢ (B ∈ V → (AFB) ∈ S)
31	19, 30	pm2.61d2 111	. 2 ⊢ (A ∈ V → (AFB) ∈ S)
32		relxp 2486	. . . . . 6 ⊢ Rel (S × S)
33		releq 2477	. . . . . . 7 ⊢ (dom F = (S × S) → (Rel dom F ↔ Rel (S × S)))
34	4, 33	ax-mp 6	. . . . . 6 ⊢ (Rel dom F ↔ Rel (S × S))
35	32, 34	mpbir 165	. . . . 5 ⊢ Rel dom F
36	35	oprprc1 3019	. . . 4 ⊢ (¬ A ∈ V → (AFB) = ∅)
37	36	eleq1d 1155	. . 3 ⊢ (¬ A ∈ V → ((AFB) ∈ S ↔ ∅ ∈ S))
38	3, 37	mpbiri 169	. 2 ⊢ (¬ A ∈ V → (AFB) ∈ S)
39	31, 38	pm2.61i 110	1 ⊢ (AFB) ∈ S

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707   × cxp 2408  dom cdm 2410  Rel wrel 2415  (class class class)co 3001

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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