Theorem enprmaplem3 6079

Description: Lemma for enprmap 6083. The converse of W is a function. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
enprmaplem3.1	⊢ W = (r ∈ (A ↑m B) ↦ (◡r “ {x}))

Assertion

Ref	Expression
enprmaplem3	⊢ ((x ≠ y ∧ A = {x, y}) → Fun ◡W)

Distinct variable groups:   A,r   B,r   x,r

Allowed substitution hints:   A(x,y)   B(x,y)   W(x,y,r)

Proof of Theorem enprmaplem3

Dummy variables p q w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brcnv 4893	. . . . . 6 ⊢ (z◡Wp ↔ pWz)
2		brcnv 4893	. . . . . 6 ⊢ (z◡Wq ↔ qWz)
3		breldm 4912	. . . . . . . . 9 ⊢ (pWz → p ∈ dom W)
4		enprmaplem3.1	. . . . . . . . . . 11 ⊢ W = (r ∈ (A ↑m B) ↦ (◡r “ {x}))
5	4	enprmaplem2 6078	. . . . . . . . . 10 ⊢ W Fn (A ↑m B)
6		fndm 5183	. . . . . . . . . 10 ⊢ (W Fn (A ↑m B) → dom W = (A ↑m B))
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ dom W = (A ↑m B)
8	3, 7	syl6eleq 2443	. . . . . . . 8 ⊢ (pWz → p ∈ (A ↑m B))
9		fnfun 5182	. . . . . . . . . . 11 ⊢ (W Fn (A ↑m B) → Fun W)
10	5, 9	ax-mp 5	. . . . . . . . . 10 ⊢ Fun W
11		funbrfv 5357	. . . . . . . . . 10 ⊢ (Fun W → (pWz → (W ‘p) = z))
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (pWz → (W ‘p) = z)
13		cnveq 4887	. . . . . . . . . . . 12 ⊢ (r = p → ◡r = ◡p)
14	13	imaeq1d 4942	. . . . . . . . . . 11 ⊢ (r = p → (◡r “ {x}) = (◡p “ {x}))
15		vex 2863	. . . . . . . . . . . . 13 ⊢ p ∈ V
16	15	cnvex 5103	. . . . . . . . . . . 12 ⊢ ◡p ∈ V
17		snex 4112	. . . . . . . . . . . 12 ⊢ {x} ∈ V
18	16, 17	imaex 4748	. . . . . . . . . . 11 ⊢ (◡p “ {x}) ∈ V
19	14, 4, 18	fvmpt 5701	. . . . . . . . . 10 ⊢ (p ∈ (A ↑m B) → (W ‘p) = (◡p “ {x}))
20	8, 19	syl 15	. . . . . . . . 9 ⊢ (pWz → (W ‘p) = (◡p “ {x}))
21	12, 20	eqtr3d 2387	. . . . . . . 8 ⊢ (pWz → z = (◡p “ {x}))
22	8, 21	jca 518	. . . . . . 7 ⊢ (pWz → (p ∈ (A ↑m B) ∧ z = (◡p “ {x})))
23		breldm 4912	. . . . . . . . 9 ⊢ (qWz → q ∈ dom W)
24	23, 7	syl6eleq 2443	. . . . . . . 8 ⊢ (qWz → q ∈ (A ↑m B))
25		funbrfv 5357	. . . . . . . . . 10 ⊢ (Fun W → (qWz → (W ‘q) = z))
26	10, 25	ax-mp 5	. . . . . . . . 9 ⊢ (qWz → (W ‘q) = z)
27		cnveq 4887	. . . . . . . . . . . 12 ⊢ (r = q → ◡r = ◡q)
28	27	imaeq1d 4942	. . . . . . . . . . 11 ⊢ (r = q → (◡r “ {x}) = (◡q “ {x}))
29		vex 2863	. . . . . . . . . . . . 13 ⊢ q ∈ V
30	29	cnvex 5103	. . . . . . . . . . . 12 ⊢ ◡q ∈ V
31	30, 17	imaex 4748	. . . . . . . . . . 11 ⊢ (◡q “ {x}) ∈ V
32	28, 4, 31	fvmpt 5701	. . . . . . . . . 10 ⊢ (q ∈ (A ↑m B) → (W ‘q) = (◡q “ {x}))
33	24, 32	syl 15	. . . . . . . . 9 ⊢ (qWz → (W ‘q) = (◡q “ {x}))
34	26, 33	eqtr3d 2387	. . . . . . . 8 ⊢ (qWz → z = (◡q “ {x}))
35	24, 34	jca 518	. . . . . . 7 ⊢ (qWz → (q ∈ (A ↑m B) ∧ z = (◡q “ {x})))
36	22, 35	anim12i 549	. . . . . 6 ⊢ ((pWz ∧ qWz) → ((p ∈ (A ↑m B) ∧ z = (◡p “ {x})) ∧ (q ∈ (A ↑m B) ∧ z = (◡q “ {x}))))
37	1, 2, 36	syl2anb 465	. . . . 5 ⊢ ((z◡Wp ∧ z◡Wq) → ((p ∈ (A ↑m B) ∧ z = (◡p “ {x})) ∧ (q ∈ (A ↑m B) ∧ z = (◡q “ {x}))))
38		elmapi 6017	. . . . . . . . 9 ⊢ (p ∈ (A ↑m B) → p:B–→A)
39		elmapi 6017	. . . . . . . . 9 ⊢ (q ∈ (A ↑m B) → q:B–→A)
40	38, 39	anim12i 549	. . . . . . . 8 ⊢ ((p ∈ (A ↑m B) ∧ q ∈ (A ↑m B)) → (p:B–→A ∧ q:B–→A))
41		eqtr2 2371	. . . . . . . 8 ⊢ ((z = (◡p “ {x}) ∧ z = (◡q “ {x})) → (◡p “ {x}) = (◡q “ {x}))
42		simprll 738	. . . . . . . . . . 11 ⊢ (((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) → p:B–→A)
43		ffn 5224	. . . . . . . . . . 11 ⊢ (p:B–→A → p Fn B)
44	42, 43	syl 15	. . . . . . . . . 10 ⊢ (((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) → p Fn B)
45		simprlr 739	. . . . . . . . . . 11 ⊢ (((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) → q:B–→A)
46		ffn 5224	. . . . . . . . . . 11 ⊢ (q:B–→A → q Fn B)
47	45, 46	syl 15	. . . . . . . . . 10 ⊢ (((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) → q Fn B)
48		ffvelrn 5416	. . . . . . . . . . . 12 ⊢ ((p:B–→A ∧ z ∈ B) → (p ‘z) ∈ A)
49	42, 48	sylan 457	. . . . . . . . . . 11 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (p ‘z) ∈ A)
50		simpllr 735	. . . . . . . . . . . . . 14 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → A = {x, y})
51	50	eleq2d 2420	. . . . . . . . . . . . 13 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) ∈ A ↔ (p ‘z) ∈ {x, y}))
52		fvex 5340	. . . . . . . . . . . . . 14 ⊢ (p ‘z) ∈ V
53	52	elpr 3752	. . . . . . . . . . . . 13 ⊢ ((p ‘z) ∈ {x, y} ↔ ((p ‘z) = x ∨ (p ‘z) = y))
54	51, 53	syl6bb 252	. . . . . . . . . . . 12 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) ∈ A ↔ ((p ‘z) = x ∨ (p ‘z) = y)))
55		simprr 733	. . . . . . . . . . . . . . 15 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ (p ‘z) = x)) → (p ‘z) = x)
56		simplrr 737	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (◡p “ {x}) = (◡q “ {x}))
57	56	eleq2d 2420	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (z ∈ (◡p “ {x}) ↔ z ∈ (◡q “ {x})))
58		eliniseg 5021	. . . . . . . . . . . . . . . . . . 19 ⊢ (z ∈ (◡p “ {x}) ↔ zpx)
59		eliniseg 5021	. . . . . . . . . . . . . . . . . . 19 ⊢ (z ∈ (◡q “ {x}) ↔ zqx)
60	57, 58, 59	3bitr3g 278	. . . . . . . . . . . . . . . . . 18 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (zpx ↔ zqx))
61	60	biimpd 198	. . . . . . . . . . . . . . . . 17 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (zpx → zqx))
62		fnbrfvb 5359	. . . . . . . . . . . . . . . . . 18 ⊢ ((p Fn B ∧ z ∈ B) → ((p ‘z) = x ↔ zpx))
63	44, 62	sylan 457	. . . . . . . . . . . . . . . . 17 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) = x ↔ zpx))
64		fnbrfvb 5359	. . . . . . . . . . . . . . . . . 18 ⊢ ((q Fn B ∧ z ∈ B) → ((q ‘z) = x ↔ zqx))
65	47, 64	sylan 457	. . . . . . . . . . . . . . . . 17 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((q ‘z) = x ↔ zqx))
66	61, 63, 65	3imtr4d 259	. . . . . . . . . . . . . . . 16 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) = x → (q ‘z) = x))
67	66	impr 602	. . . . . . . . . . . . . . 15 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ (p ‘z) = x)) → (q ‘z) = x)
68	55, 67	eqtr4d 2388	. . . . . . . . . . . . . 14 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ (p ‘z) = x)) → (p ‘z) = (q ‘z))
69	68	expr 598	. . . . . . . . . . . . 13 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) = x → (p ‘z) = (q ‘z)))
70		simprr 733	. . . . . . . . . . . . . . 15 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ (p ‘z) = y)) → (p ‘z) = y)
71		simplll 734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → x ≠ y)
72	71	neneqd 2533	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → ¬ x = y)
73	42	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → p:B–→A)
74		ffun 5226	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (p:B–→A → Fun p)
75		fununiq 5518	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun p ∧ zpx ∧ zpy) → x = y)
76	75	3expib 1154	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Fun p → ((zpx ∧ zpy) → x = y))
77	76	ancomsd 440	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Fun p → ((zpy ∧ zpx) → x = y))
78	73, 74, 77	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((zpy ∧ zpx) → x = y))
79	78	exp3a 425	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (zpy → (zpx → x = y)))
80	79	impr 602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (zpx → x = y))
81	72, 80	mtod 168	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → ¬ zpx)
82	81	expr 598	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (zpy → ¬ zpx))
83	60	biimprd 214	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (zqx → zpx))
84	82, 83	nsyld 132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (zpy → ¬ zqx))
85	84	impr 602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → ¬ zqx)
86		simprl 732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → z ∈ B)
87	45	adantr 451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → q:B–→A)
88		fdm 5227	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (q:B–→A → dom q = B)
89	87, 88	syl 15	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → dom q = B)
90	86, 89	eleqtrrd 2430	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → z ∈ dom q)
91		eldm 4899	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (z ∈ dom q ↔ ∃w zqw)
92		brelrn 4961	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (zqw → w ∈ ran q)
93		frn 5229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (q:B–→A → ran q ⊆ A)
94	87, 93	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → ran q ⊆ A)
95	94	sseld 3273	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (w ∈ ran q → w ∈ A))
96	92, 95	syl5 28	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (zqw → w ∈ A))
97		simpllr 735	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → A = {x, y})
98	97	eleq2d 2420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (w ∈ A ↔ w ∈ {x, y}))
99		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ w ∈ V
100	99	elpr 3752	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (w ∈ {x, y} ↔ (w = x ∨ w = y))
101		breq2 4644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (w = x → (zqw ↔ zqx))
102	101	biimpcd 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (zqw → (w = x → zqx))
103		breq2 4644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (w = y → (zqw ↔ zqy))
104	103	biimpcd 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (zqw → (w = y → zqy))
105	102, 104	orim12d 811	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (zqw → ((w = x ∨ w = y) → (zqx ∨ zqy)))
106	105	com12 27	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((w = x ∨ w = y) → (zqw → (zqx ∨ zqy)))
107	100, 106	sylbi 187	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (w ∈ {x, y} → (zqw → (zqx ∨ zqy)))
108	98, 107	syl6bi 219	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (w ∈ A → (zqw → (zqx ∨ zqy))))
109	108	com23 72	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (zqw → (w ∈ A → (zqx ∨ zqy))))
110	96, 109	mpdd 36	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (zqw → (zqx ∨ zqy)))
111	110	exlimdv 1636	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (∃w zqw → (zqx ∨ zqy)))
112	91, 111	syl5bi 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (z ∈ dom q → (zqx ∨ zqy)))
113	90, 112	mpd 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → (zqx ∨ zqy))
114		orel1 371	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ zqx → ((zqx ∨ zqy) → zqy))
115	85, 113, 114	sylc 56	. . . . . . . . . . . . . . . . . 18 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ zpy)) → zqy)
116	115	expr 598	. . . . . . . . . . . . . . . . 17 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (zpy → zqy))
117		fnbrfvb 5359	. . . . . . . . . . . . . . . . . 18 ⊢ ((p Fn B ∧ z ∈ B) → ((p ‘z) = y ↔ zpy))
118	44, 117	sylan 457	. . . . . . . . . . . . . . . . 17 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) = y ↔ zpy))
119		fnbrfvb 5359	. . . . . . . . . . . . . . . . . 18 ⊢ ((q Fn B ∧ z ∈ B) → ((q ‘z) = y ↔ zqy))
120	47, 119	sylan 457	. . . . . . . . . . . . . . . . 17 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((q ‘z) = y ↔ zqy))
121	116, 118, 120	3imtr4d 259	. . . . . . . . . . . . . . . 16 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) = y → (q ‘z) = y))
122	121	impr 602	. . . . . . . . . . . . . . 15 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ (p ‘z) = y)) → (q ‘z) = y)
123	70, 122	eqtr4d 2388	. . . . . . . . . . . . . 14 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ (z ∈ B ∧ (p ‘z) = y)) → (p ‘z) = (q ‘z))
124	123	expr 598	. . . . . . . . . . . . 13 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) = y → (p ‘z) = (q ‘z)))
125	69, 124	jaod 369	. . . . . . . . . . . 12 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (((p ‘z) = x ∨ (p ‘z) = y) → (p ‘z) = (q ‘z)))
126	54, 125	sylbid 206	. . . . . . . . . . 11 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → ((p ‘z) ∈ A → (p ‘z) = (q ‘z)))
127	49, 126	mpd 14	. . . . . . . . . 10 ⊢ ((((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) ∧ z ∈ B) → (p ‘z) = (q ‘z))
128	44, 47, 127	eqfnfvd 5396	. . . . . . . . 9 ⊢ (((x ≠ y ∧ A = {x, y}) ∧ ((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x}))) → p = q)
129	128	expcom 424	. . . . . . . 8 ⊢ (((p:B–→A ∧ q:B–→A) ∧ (◡p “ {x}) = (◡q “ {x})) → ((x ≠ y ∧ A = {x, y}) → p = q))
130	40, 41, 129	syl2an 463	. . . . . . 7 ⊢ (((p ∈ (A ↑m B) ∧ q ∈ (A ↑m B)) ∧ (z = (◡p “ {x}) ∧ z = (◡q “ {x}))) → ((x ≠ y ∧ A = {x, y}) → p = q))
131	130	an4s 799	. . . . . 6 ⊢ (((p ∈ (A ↑m B) ∧ z = (◡p “ {x})) ∧ (q ∈ (A ↑m B) ∧ z = (◡q “ {x}))) → ((x ≠ y ∧ A = {x, y}) → p = q))
132	131	com12 27	. . . . 5 ⊢ ((x ≠ y ∧ A = {x, y}) → (((p ∈ (A ↑m B) ∧ z = (◡p “ {x})) ∧ (q ∈ (A ↑m B) ∧ z = (◡q “ {x}))) → p = q))
133	37, 132	syl5 28	. . . 4 ⊢ ((x ≠ y ∧ A = {x, y}) → ((z◡Wp ∧ z◡Wq) → p = q))
134	133	alrimiv 1631	. . 3 ⊢ ((x ≠ y ∧ A = {x, y}) → ∀q((z◡Wp ∧ z◡Wq) → p = q))
135	134	alrimivv 1632	. 2 ⊢ ((x ≠ y ∧ A = {x, y}) → ∀z∀p∀q((z◡Wp ∧ z◡Wq) → p = q))
136		dffun2 5120	. 2 ⊢ (Fun ◡W ↔ ∀z∀p∀q((z◡Wp ∧ z◡Wq) → p = q))
137	135, 136	sylibr 203	1 ⊢ ((x ≠ y ∧ A = {x, y}) → Fun ◡W)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517   ⊆ wss 3258  {csn 3738  {cpr 3739   class class class wbr 4640   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  –→wf 4778   ‘cfv 4782  (class class class)co 5526   ↦ cmpt 5652   ↑_m cmap 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002

This theorem is referenced by:  enprmap  6083