Theorem unxpdomlem 5995

Description: Lemma for unxpdom 5996.

Hypotheses

Ref	Expression
unxpdomlem.1	|- A e. _V
unxpdomlem.2	|- B e. _V

Assertion

Ref	Expression
unxpdomlem	|- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))

Proof of Theorem unxpdomlem

Step	Hyp	Ref	Expression
1		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . 21 |- (x = f -> (x = w <-> f = w))
2	1	ifbid 2996	. . . . . . . . . . . . . . . . . . . 20 |- (x = f -> if(x = w, if(y = v, w, y), if(y = v, v, x)) = if(f = w, if(y = v, w, y), if(y = v, v, x)))
3		ifeq2 2987	. . . . . . . . . . . . . . . . . . . . 21 |- (x = f -> if(y = v, v, x) = if(y = v, v, f))
4	3	ifeq2d 2994	. . . . . . . . . . . . . . . . . . . 20 |- (x = f -> if(f = w, if(y = v, w, y), if(y = v, v, x)) = if(f = w, if(y = v, w, y), if(y = v, v, f)))
5	2, 4	eqtrd 1925	. . . . . . . . . . . . . . . . . . 19 |- (x = f -> if(x = w, if(y = v, w, y), if(y = v, v, x)) = if(f = w, if(y = v, w, y), if(y = v, v, f)))
6	5	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- (x = f -> (if(x = w, if(y = v, w, y), if(y = v, v, x)) = f <-> if(f = w, if(y = v, w, y), if(y = v, v, f)) = f))
7		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = v -> (y = v <-> v = v))
8	7	ifbid 2996	. . . . . . . . . . . . . . . . . . . . 21 |- (y = v -> if(y = v, w, y) = if(v = v, w, y))
9		ifeq2 2987	. . . . . . . . . . . . . . . . . . . . 21 |- (y = v -> if(v = v, w, y) = if(v = v, w, v))
10	8, 9	eqtrd 1925	. . . . . . . . . . . . . . . . . . . 20 |- (y = v -> if(y = v, w, y) = if(v = v, w, v))
11	7	ifbid 2996	. . . . . . . . . . . . . . . . . . . 20 |- (y = v -> if(y = v, v, f) = if(v = v, v, f))
12		ifeq12 2992	. . . . . . . . . . . . . . . . . . . 20 |- ((if(y = v, w, y) = if(v = v, w, v) /\ if(y = v, v, f) = if(v = v, v, f)) -> if(f = w, if(y = v, w, y), if(y = v, v, f)) = if(f = w, if(v = v, w, v), if(v = v, v, f)))
13	10, 11, 12	syl11anc 524	. . . . . . . . . . . . . . . . . . 19 |- (y = v -> if(f = w, if(y = v, w, y), if(y = v, v, f)) = if(f = w, if(v = v, w, v), if(v = v, v, f)))
14	13	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- (y = v -> (if(f = w, if(y = v, w, y), if(y = v, v, f)) = f <-> if(f = w, if(v = v, w, v), if(v = v, v, f)) = f))
15	6, 14	rcla42ev 2385	. . . . . . . . . . . . . . . . 17 |- ((f e. A /\ v e. B /\ if(f = w, if(v = v, w, v), if(v = v, v, f)) = f) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
16		eqid 1884	. . . . . . . . . . . . . . . . . . 19 |- v = v
17		iftrue 2989	. . . . . . . . . . . . . . . . . . 19 |- (v = v -> if(v = v, w, v) = w)
18	16, 17	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- if(v = v, w, v) = w
19		iftrue 2989	. . . . . . . . . . . . . . . . . 18 |- (f = w -> if(f = w, if(v = v, w, v), if(v = v, v, f)) = if(v = v, w, v))
20		id 73	. . . . . . . . . . . . . . . . . 18 |- (f = w -> f = w)
21	18, 19, 20	3eqtr4a 1954	. . . . . . . . . . . . . . . . 17 |- (f = w -> if(f = w, if(v = v, w, v), if(v = v, v, f)) = f)
22	15, 21	syl3an3 1132	. . . . . . . . . . . . . . . 16 |- ((f e. A /\ v e. B /\ f = w) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
23	22	3exp 1066	. . . . . . . . . . . . . . 15 |- (f e. A -> (v e. B -> (f = w -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
24	23	com3l 38	. . . . . . . . . . . . . 14 |- (v e. B -> (f = w -> (f e. A -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
25	24	ad2antlr 441	. . . . . . . . . . . . 13 |- (((t e. B /\ v e. B) /\ -. t = v) -> (f = w -> (f e. A -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
26		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = t -> (y = v <-> t = v))
27	26	ifbid 2996	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = t -> if(y = v, w, y) = if(t = v, w, y))
28		ifeq2 2987	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = t -> if(t = v, w, y) = if(t = v, w, t))
29	27, 28	eqtrd 1925	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = t -> if(y = v, w, y) = if(t = v, w, t))
30	26	ifbid 2996	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = t -> if(y = v, v, f) = if(t = v, v, f))
31		ifeq12 2992	. . . . . . . . . . . . . . . . . . . . . 22 |- ((if(y = v, w, y) = if(t = v, w, t) /\ if(y = v, v, f) = if(t = v, v, f)) -> if(f = w, if(y = v, w, y), if(y = v, v, f)) = if(f = w, if(t = v, w, t), if(t = v, v, f)))
32	29, 30, 31	syl11anc 524	. . . . . . . . . . . . . . . . . . . . 21 |- (y = t -> if(f = w, if(y = v, w, y), if(y = v, v, f)) = if(f = w, if(t = v, w, t), if(t = v, v, f)))
33	32	eqeq1d 1892	. . . . . . . . . . . . . . . . . . . 20 |- (y = t -> (if(f = w, if(y = v, w, y), if(y = v, v, f)) = f <-> if(f = w, if(t = v, w, t), if(t = v, v, f)) = f))
34	6, 33	rcla42ev 2385	. . . . . . . . . . . . . . . . . . 19 |- ((f e. A /\ t e. B /\ if(f = w, if(t = v, w, t), if(t = v, v, f)) = f) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
35		iffalse 2991	. . . . . . . . . . . . . . . . . . . 20 |- (-. f = w -> if(f = w, if(t = v, w, t), if(t = v, v, f)) = if(t = v, v, f))
36		iffalse 2991	. . . . . . . . . . . . . . . . . . . 20 |- (-. t = v -> if(t = v, v, f) = f)
37	35, 36	sylan9eqr 1951	. . . . . . . . . . . . . . . . . . 19 |- ((-. t = v /\ -. f = w) -> if(f = w, if(t = v, w, t), if(t = v, v, f)) = f)
38	34, 37	syl3an3 1132	. . . . . . . . . . . . . . . . . 18 |- ((f e. A /\ t e. B /\ (-. t = v /\ -. f = w)) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
39	38	3exp 1066	. . . . . . . . . . . . . . . . 17 |- (f e. A -> (t e. B -> ((-. t = v /\ -. f = w) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
40	39	exp4a 409	. . . . . . . . . . . . . . . 16 |- (f e. A -> (t e. B -> (-. t = v -> (-. f = w -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))))
41	40	imp3a 388	. . . . . . . . . . . . . . 15 |- (f e. A -> ((t e. B /\ -. t = v) -> (-. f = w -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
42	41	com3l 38	. . . . . . . . . . . . . 14 |- ((t e. B /\ -. t = v) -> (-. f = w -> (f e. A -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
43	42	adantlr 429	. . . . . . . . . . . . 13 |- (((t e. B /\ v e. B) /\ -. t = v) -> (-. f = w -> (f e. A -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
44	25, 43	pm2.61d 141	. . . . . . . . . . . 12 |- (((t e. B /\ v e. B) /\ -. t = v) -> (f e. A -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))
45	44	adantl 424	. . . . . . . . . . 11 |- ((((u e. A /\ w e. A) /\ -. u = w) /\ ((t e. B /\ v e. B) /\ -. t = v)) -> (f e. A -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))
46		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x = u -> (x = w <-> u = w))
47	46	ifbid 2996	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = u -> if(x = w, if(y = v, w, y), if(y = v, v, x)) = if(u = w, if(y = v, w, y), if(y = v, v, x)))
48		ifeq2 2987	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x = u -> if(y = v, v, x) = if(y = v, v, u))
49	48	ifeq2d 2994	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = u -> if(u = w, if(y = v, w, y), if(y = v, v, x)) = if(u = w, if(y = v, w, y), if(y = v, v, u)))
50	47, 49	eqtrd 1925	. . . . . . . . . . . . . . . . . . . . 21 |- (x = u -> if(x = w, if(y = v, w, y), if(y = v, v, x)) = if(u = w, if(y = v, w, y), if(y = v, v, u)))
51	50	eqeq1d 1892	. . . . . . . . . . . . . . . . . . . 20 |- (x = u -> (if(x = w, if(y = v, w, y), if(y = v, v, x)) = f <-> if(u = w, if(y = v, w, y), if(y = v, v, u)) = f))
52		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = f -> (y = v <-> f = v))
53	52	ifbid 2996	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = f -> if(y = v, w, y) = if(f = v, w, y))
54		ifeq2 2987	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = f -> if(f = v, w, y) = if(f = v, w, f))
55	53, 54	eqtrd 1925	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = f -> if(y = v, w, y) = if(f = v, w, f))
56	52	ifbid 2996	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = f -> if(y = v, v, u) = if(f = v, v, u))
57		ifeq12 2992	. . . . . . . . . . . . . . . . . . . . . 22 |- ((if(y = v, w, y) = if(f = v, w, f) /\ if(y = v, v, u) = if(f = v, v, u)) -> if(u = w, if(y = v, w, y), if(y = v, v, u)) = if(u = w, if(f = v, w, f), if(f = v, v, u)))
58	55, 56, 57	syl11anc 524	. . . . . . . . . . . . . . . . . . . . 21 |- (y = f -> if(u = w, if(y = v, w, y), if(y = v, v, u)) = if(u = w, if(f = v, w, f), if(f = v, v, u)))
59	58	eqeq1d 1892	. . . . . . . . . . . . . . . . . . . 20 |- (y = f -> (if(u = w, if(y = v, w, y), if(y = v, v, u)) = f <-> if(u = w, if(f = v, w, f), if(f = v, v, u)) = f))
60	51, 59	rcla42ev 2385	. . . . . . . . . . . . . . . . . . 19 |- ((u e. A /\ f e. B /\ if(u = w, if(f = v, w, f), if(f = v, v, u)) = f) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
61		iffalse 2991	. . . . . . . . . . . . . . . . . . . 20 |- (-. u = w -> if(u = w, if(f = v, w, f), if(f = v, v, u)) = if(f = v, v, u))
62		iftrue 2989	. . . . . . . . . . . . . . . . . . . . 21 |- (f = v -> if(f = v, v, u) = v)
63		equcomi 1487	. . . . . . . . . . . . . . . . . . . . 21 |- (f = v -> v = f)
64	62, 63	eqtrd 1925	. . . . . . . . . . . . . . . . . . . 20 |- (f = v -> if(f = v, v, u) = f)
65	61, 64	sylan9eqr 1951	. . . . . . . . . . . . . . . . . . 19 |- ((f = v /\ -. u = w) -> if(u = w, if(f = v, w, f), if(f = v, v, u)) = f)
66	60, 65	syl3an3 1132	. . . . . . . . . . . . . . . . . 18 |- ((u e. A /\ f e. B /\ (f = v /\ -. u = w)) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
67	66	3exp 1066	. . . . . . . . . . . . . . . . 17 |- (u e. A -> (f e. B -> ((f = v /\ -. u = w) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
68	67	exp4a 409	. . . . . . . . . . . . . . . 16 |- (u e. A -> (f e. B -> (f = v -> (-. u = w -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))))
69	68	com24 41	. . . . . . . . . . . . . . 15 |- (u e. A -> (-. u = w -> (f = v -> (f e. B -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))))
70	69	imp 377	. . . . . . . . . . . . . 14 |- ((u e. A /\ -. u = w) -> (f = v -> (f e. B -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
71	70	adantlr 429	. . . . . . . . . . . . 13 |- (((u e. A /\ w e. A) /\ -. u = w) -> (f = v -> (f e. B -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
72		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . 21 |- (x = w -> (x = w <-> w = w))
73	72	ifbid 2996	. . . . . . . . . . . . . . . . . . . 20 |- (x = w -> if(x = w, if(y = v, w, y), if(y = v, v, x)) = if(w = w, if(y = v, w, y), if(y = v, v, x)))
74		ifeq2 2987	. . . . . . . . . . . . . . . . . . . . 21 |- (x = w -> if(y = v, v, x) = if(y = v, v, w))
75	74	ifeq2d 2994	. . . . . . . . . . . . . . . . . . . 20 |- (x = w -> if(w = w, if(y = v, w, y), if(y = v, v, x)) = if(w = w, if(y = v, w, y), if(y = v, v, w)))
76	73, 75	eqtrd 1925	. . . . . . . . . . . . . . . . . . 19 |- (x = w -> if(x = w, if(y = v, w, y), if(y = v, v, x)) = if(w = w, if(y = v, w, y), if(y = v, v, w)))
77	76	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- (x = w -> (if(x = w, if(y = v, w, y), if(y = v, v, x)) = f <-> if(w = w, if(y = v, w, y), if(y = v, v, w)) = f))
78	52	ifbid 2996	. . . . . . . . . . . . . . . . . . . 20 |- (y = f -> if(y = v, v, w) = if(f = v, v, w))
79		ifeq12 2992	. . . . . . . . . . . . . . . . . . . 20 |- ((if(y = v, w, y) = if(f = v, w, f) /\ if(y = v, v, w) = if(f = v, v, w)) -> if(w = w, if(y = v, w, y), if(y = v, v, w)) = if(w = w, if(f = v, w, f), if(f = v, v, w)))
80	55, 78, 79	syl11anc 524	. . . . . . . . . . . . . . . . . . 19 |- (y = f -> if(w = w, if(y = v, w, y), if(y = v, v, w)) = if(w = w, if(f = v, w, f), if(f = v, v, w)))
81	80	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- (y = f -> (if(w = w, if(y = v, w, y), if(y = v, v, w)) = f <-> if(w = w, if(f = v, w, f), if(f = v, v, w)) = f))
82	77, 81	rcla42ev 2385	. . . . . . . . . . . . . . . . 17 |- ((w e. A /\ f e. B /\ if(w = w, if(f = v, w, f), if(f = v, v, w)) = f) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
83		iffalse 2991	. . . . . . . . . . . . . . . . . 18 |- (-. f = v -> if(f = v, w, f) = f)
84		eqid 1884	. . . . . . . . . . . . . . . . . . 19 |- w = w
85		iftrue 2989	. . . . . . . . . . . . . . . . . . 19 |- (w = w -> if(w = w, if(f = v, w, f), if(f = v, v, w)) = if(f = v, w, f))
86	84, 85	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- if(w = w, if(f = v, w, f), if(f = v, v, w)) = if(f = v, w, f)
87	83, 86	syl5eq 1940	. . . . . . . . . . . . . . . . 17 |- (-. f = v -> if(w = w, if(f = v, w, f), if(f = v, v, w)) = f)
88	82, 87	syl3an3 1132	. . . . . . . . . . . . . . . 16 |- ((w e. A /\ f e. B /\ -. f = v) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)
89	88	3exp 1066	. . . . . . . . . . . . . . 15 |- (w e. A -> (f e. B -> (-. f = v -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
90	89	com23 36	. . . . . . . . . . . . . 14 |- (w e. A -> (-. f = v -> (f e. B -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
91	90	ad2antlr 441	. . . . . . . . . . . . 13 |- (((u e. A /\ w e. A) /\ -. u = w) -> (-. f = v -> (f e. B -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f)))
92	71, 91	pm2.61d 141	. . . . . . . . . . . 12 |- (((u e. A /\ w e. A) /\ -. u = w) -> (f e. B -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))
93	92	adantr 425	. . . . . . . . . . 11 |- ((((u e. A /\ w e. A) /\ -. u = w) /\ ((t e. B /\ v e. B) /\ -. t = v)) -> (f e. B -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))
94	45, 93	jaod 469	. . . . . . . . . 10 |- ((((u e. A /\ w e. A) /\ -. u = w) /\ ((t e. B /\ v e. B) /\ -. t = v)) -> ((f e. A \/ f e. B) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))
95		elun 2741	. . . . . . . . . 10 |- (f e. (A u. B) <-> (f e. A \/ f e. B))
96	94, 95	syl5ib 223	. . . . . . . . 9 |- ((((u e. A /\ w e. A) /\ -. u = w) /\ ((t e. B /\ v e. B) /\ -. t = v)) -> (f e. (A u. B) -> E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f))
97		eqcom 1886	. . . . . . . . . . 11 |- (if(x = w, if(y = v, w, y), if(y = v, v, x)) = f <-> f = if(x = w, if(y = v, w, y), if(y = v, v, x)))
98	97	2rexbii 2130	. . . . . . . . . 10 |- (E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f <-> E.x e. A E.y e. B f = if(x = w, if(y = v, w, y), if(y = v, v, x)))
99		visset 2295	. . . . . . . . . . . . 13 |- w e. _V
100		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
101	99, 100	ifex 3031	. . . . . . . . . . . 12 |- if(y = v, w, y) e. _V
102		visset 2295	. . . . . . . . . . . . 13 |- v e. _V
103		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
104	102, 103	ifex 3031	. . . . . . . . . . . 12 |- if(y = v, v, x) e. _V
105	101, 104	ifex 3031	. . . . . . . . . . 11 |- if(x = w, if(y = v, w, y), if(y = v, v, x)) e. _V
106		eqid 1884	. . . . . . . . . . 11 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))}
107	105, 106	elrnoprab 5067	. . . . . . . . . 10 |- (f e. ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} <-> E.x e. A E.y e. B f = if(x = w, if(y = v, w, y), if(y = v, v, x)))
108	98, 107	bitr4i 193	. . . . . . . . 9 |- (E.x e. A E.y e. B if(x = w, if(y = v, w, y), if(y = v, v, x)) = f <-> f e. ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))})
109	96, 108	syl6ib 229	. . . . . . . 8 |- ((((u e. A /\ w e. A) /\ -. u = w) /\ ((t e. B /\ v e. B) /\ -. t = v)) -> (f e. (A u. B) -> f e. ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))}))
110	109	ssrdv 2622	. . . . . . 7 |- ((((u e. A /\ w e. A) /\ -. u = w) /\ ((t e. B /\ v e. B) /\ -. t = v)) -> (A u. B) C_ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))})
111		unxpdomlem.1	. . . . . . . . 9 |- A e. _V
112		unxpdomlem.2	. . . . . . . . 9 |- B e. _V
113	111, 112	unex 3796	. . . . . . . 8 |- (A u. B) e. _V
114		ssdomg 5467	. . . . . . . 8 |- ((A u. B) e. _V -> ((A u. B) C_ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} -> (A u. B) ~<_ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))}))
115	113, 114	ax-mp 7	. . . . . . 7 |- ((A u. B) C_ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} -> (A u. B) ~<_ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))})
116	111, 112	xpex 4096	. . . . . . . . 9 |- (A X. B) e. _V
117	105, 106	fnoprab2 5064	. . . . . . . . 9 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} Fn (A X. B)
118		fnrndomg 5969	. . . . . . . . 9 |- ((A X. B) e. _V -> ({<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} Fn (A X. B) -> ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} ~<_ (A X. B)))
119	116, 117, 118	mp2 54	. . . . . . . 8 |- ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} ~<_ (A X. B)
120		domtr 5474	. . . . . . . 8 |- (((A u. B) ~<_ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} /\ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} ~<_ (A X. B)) -> (A u. B) ~<_ (A X. B))
121	119, 120	mpan2 760	. . . . . . 7 |- ((A u. B) ~<_ ran {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = if(x = w, if(y = v, w, y), if(y = v, v, x)))} -> (A u. B) ~<_ (A X. B))
122	110, 115, 121	3syl 24	. . . . . 6 |- ((((u e. A /\ w e. A) /\ -. u = w) /\ ((t e. B /\ v e. B) /\ -. t = v)) -> (A u. B) ~<_ (A X. B))
123	122	exp43 415	. . . . 5 |- ((u e. A /\ w e. A) -> (-. u = w -> ((t e. B /\ v e. B) -> (-. t = v -> (A u. B) ~<_ (A X. B)))))
124	123	r19.23aivv 2217	. . . 4 |- (E.u e. A E.w e. A -. u = w -> ((t e. B /\ v e. B) -> (-. t = v -> (A u. B) ~<_ (A X. B))))
125	124	r19.23advv 2218	. . 3 |- (E.u e. A E.w e. A -. u = w -> (E.t e. B E.v e. B -. t = v -> (A u. B) ~<_ (A X. B)))
126	125	imp 377	. 2 |- ((E.u e. A E.w e. A -. u = w /\ E.t e. B E.v e. B -. t = v) -> (A u. B) ~<_ (A X. B))
127		1onn 5310	. . . . 5 |- 1o e. om
128		sucdom 5994	. . . . 5 |- ((1o e. om /\ A e. _V) -> (1o ~< A <-> suc 1o ~<_ A))
129	127, 111, 128	mp2an 761	. . . 4 |- (1o ~< A <-> suc 1o ~<_ A)
130		df-2o 5178	. . . . 5 |- 2o = suc 1o
131	130	breq1i 3345	. . . 4 |- (2o ~<_ A <-> suc 1o ~<_ A)
132	129, 131	bitr4i 193	. . 3 |- (1o ~< A <-> 2o ~<_ A)
133	111	2dom 5486	. . 3 |- (2o ~<_ A -> E.u e. A E.w e. A -. u = w)
134	132, 133	sylbi 216	. 2 |- (1o ~< A -> E.u e. A E.w e. A -. u = w)
135		sucdom 5994	. . . . 5 |- ((1o e. om /\ B e. _V) -> (1o ~< B <-> suc 1o ~<_ B))
136	127, 112, 135	mp2an 761	. . . 4 |- (1o ~< B <-> suc 1o ~<_ B)
137	130	breq1i 3345	. . . 4 |- (2o ~<_ B <-> suc 1o ~<_ B)
138	136, 137	bitr4i 193	. . 3 |- (1o ~< B <-> 2o ~<_ B)
139	112	2dom 5486	. . 3 |- (2o ~<_ B -> E.t e. B E.v e. B -. t = v)
140	138, 139	sylbi 216	. 2 |- (1o ~< B -> E.t e. B E.v e. B -. t = v)
141	126, 134, 140	syl2an 503	1 |- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  ifcif 2982   class class class wbr 3338  suc csuc 3659  omcom 3949   X. cxp 3984  ran crn 3987   Fn wfn 3993  {copab2 4885  1oc1o 5172  2oc2o 5173   ~<_ cdom 5424   ~< csdm 5425

This theorem is referenced by:  unxpdom 5996

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-2o 5178  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862

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