Theorem ndmoprcl 4047

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. It is is dependent on our particular definitions of operation value, function value, and ordered pair.

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 groups:   x,A   x,B   x,F   x,S

Proof of Theorem ndmoprcl

Step	Hyp	Ref	Expression
1		oprprc2 3987	. . . . 5 ⊢ (¬ B ∈ V → (AFB) = (AFA))
2	1	eleq1d 1537	. . . 4 ⊢ (¬ B ∈ V → ((AFB) ∈ S ↔ (AFA) ∈ S))
3		ndmoprcl.1	. . . . . . . 8 ⊢ dom F = (S × S)
4	3	ndmoprgOLD 4046	. . . . . . 7 ⊢ ((A ∈ V ⋀ ¬ (A ∈ S ⋀ A ∈ S)) → (AFA) = ∅)
5		ndmoprcl.3	. . . . . . 7 ⊢ ∅ ∈ S
6	4, 5	syl6eqel 1553	. . . . . 6 ⊢ ((A ∈ V ⋀ ¬ (A ∈ S ⋀ A ∈ S)) → (AFA) ∈ S)
7	6	ex 373	. . . . 5 ⊢ (A ∈ V → (¬ (A ∈ S ⋀ A ∈ S) → (AFA) ∈ S))
8		opreq2 3971	. . . . . . . . 9 ⊢ (x = A → (AFx) = (AFA))
9	8	eleq1d 1537	. . . . . . . 8 ⊢ (x = A → ((AFx) ∈ S ↔ (AFA) ∈ S))
10	9	imbi2d 611	. . . . . . 7 ⊢ (x = A → ((A ∈ S → (AFx) ∈ S) ↔ (A ∈ S → (AFA) ∈ S)))
11		ndmoprcl.2	. . . . . . . 8 ⊢ ((A ∈ S ⋀ x ∈ S) → (AFx) ∈ S)
12	11	expcom 374	. . . . . . 7 ⊢ (x ∈ S → (A ∈ S → (AFx) ∈ S))
13	10, 12	vtoclga 1848	. . . . . 6 ⊢ (A ∈ S → (A ∈ S → (AFA) ∈ S))
14	13	imp 350	. . . . 5 ⊢ ((A ∈ S ⋀ A ∈ S) → (AFA) ∈ S)
15	7, 14	pm2.61d2 129	. . . 4 ⊢ (A ∈ V → (AFA) ∈ S)
16	2, 15	syl5cbir 211	. . 3 ⊢ (A ∈ V → (¬ B ∈ V → (AFB) ∈ S))
17	3	ndmoprgOLD 4046	. . . . . 6 ⊢ ((B ∈ V ⋀ ¬ (A ∈ S ⋀ B ∈ S)) → (AFB) = ∅)
18	17, 5	syl6eqel 1553	. . . . 5 ⊢ ((B ∈ V ⋀ ¬ (A ∈ S ⋀ B ∈ S)) → (AFB) ∈ S)
19	18	ex 373	. . . 4 ⊢ (B ∈ V → (¬ (A ∈ S ⋀ B ∈ S) → (AFB) ∈ S))
20		opreq2 3971	. . . . . . . 8 ⊢ (x = B → (AFx) = (AFB))
21	20	eleq1d 1537	. . . . . . 7 ⊢ (x = B → ((AFx) ∈ S ↔ (AFB) ∈ S))
22	21	imbi2d 611	. . . . . 6 ⊢ (x = B → ((A ∈ S → (AFx) ∈ S) ↔ (A ∈ S → (AFB) ∈ S)))
23	22, 12	vtoclga 1848	. . . . 5 ⊢ (B ∈ S → (A ∈ S → (AFB) ∈ S))
24	23	impcom 351	. . . 4 ⊢ ((A ∈ S ⋀ B ∈ S) → (AFB) ∈ S)
25	19, 24	pm2.61d2 129	. . 3 ⊢ (B ∈ V → (AFB) ∈ S)
26	16, 25	pm2.61d2 129	. 2 ⊢ (A ∈ V → (AFB) ∈ S)
27		relxp 3255	. . . . 5 ⊢ Rel (S × S)
28	3	releqi 3244	. . . . 5 ⊢ (Rel dom F ↔ Rel (S × S))
29	27, 28	mpbir 190	. . . 4 ⊢ Rel dom F
30	29	oprprc1 3986	. . 3 ⊢ (¬ A ∈ V → (AFB) = ∅)
31	30, 5	syl6eqel 1553	. 2 ⊢ (¬ A ∈ V → (AFB) ∈ S)
32	26, 31	pm2.61i 126	1 ⊢ (AFB) ∈ S

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   = wceq 954   ∈ wcel 956  Vcvv 1807  ∅c0 2276   × cxp 3168  dom cdm 3170  Rel wrel 3175  (class class class)co 3965

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-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 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1808  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-op 2412  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3967

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