Theorem 2mpt20 6882

Description: If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.)

Hypotheses

Ref	Expression
2mpt20.o	|- O = ( x e. A , y e. B |-> E )
2mpt20.u	|- ( ( X e. A /\ Y e. B ) -> ( X O Y ) = ( s e. C , t e. D |-> F ) )

Assertion

Ref	Expression
2mpt20	|- ( -. ( ( X e. A /\ Y e. B ) /\ ( S e. C /\ T e. D ) ) -> ( S ( X O Y ) T ) = (/) )

Distinct variable groups:   x, A, y   x, B, y   C, s, t   D, s, t

Allowed substitution hints:   A( t, s)   B( t, s)   C( x, y)   D( x, y)   S( x, y, t, s)   T( x, y, t, s)   E( x, y, t, s)   F( x, y, t, s)   O( x, y, t, s)   X( x, y, t, s)   Y( x, y, t, s)

Proof of Theorem 2mpt20

Step	Hyp	Ref	Expression
1		ianor 509	. 2 |- ( -. ( ( X e. A /\ Y e. B ) /\ ( S e. C /\ T e. D ) ) <-> ( -. ( X e. A /\ Y e. B ) \/ -. ( S e. C /\ T e. D ) ) )
2		2mpt20.o	. . . . . 6 |- O = ( x e. A , y e. B |-> E )
3	2	mpt2ndm0 6875	. . . . 5 |- ( -. ( X e. A /\ Y e. B ) -> ( X O Y ) = (/) )
4	3	oveqd 6667	. . . 4 |- ( -. ( X e. A /\ Y e. B ) -> ( S ( X O Y ) T ) = ( S (/) T ) )
5		0ov 6682	. . . 4 |- ( S (/) T ) = (/)
6	4, 5	syl6eq 2672	. . 3 |- ( -. ( X e. A /\ Y e. B ) -> ( S ( X O Y ) T ) = (/) )
7		notnotb 304	. . . 4 |- ( ( X e. A /\ Y e. B ) <-> -. -. ( X e. A /\ Y e. B ) )
8		2mpt20.u	. . . . . . 7 |- ( ( X e. A /\ Y e. B ) -> ( X O Y ) = ( s e. C , t e. D |-> F ) )
9	8	adantr 481	. . . . . 6 |- ( ( ( X e. A /\ Y e. B ) /\ -. ( S e. C /\ T e. D ) ) -> ( X O Y ) = ( s e. C , t e. D |-> F ) )
10	9	oveqd 6667	. . . . 5 |- ( ( ( X e. A /\ Y e. B ) /\ -. ( S e. C /\ T e. D ) ) -> ( S ( X O Y ) T ) = ( S ( s e. C , t e. D |-> F ) T ) )
11		eqid 2622	. . . . . . 7 |- ( s e. C , t e. D |-> F ) = ( s e. C , t e. D |-> F )
12	11	mpt2ndm0 6875	. . . . . 6 |- ( -. ( S e. C /\ T e. D ) -> ( S ( s e. C , t e. D |-> F ) T ) = (/) )
13	12	adantl 482	. . . . 5 |- ( ( ( X e. A /\ Y e. B ) /\ -. ( S e. C /\ T e. D ) ) -> ( S ( s e. C , t e. D |-> F ) T ) = (/) )
14	10, 13	eqtrd 2656	. . . 4 |- ( ( ( X e. A /\ Y e. B ) /\ -. ( S e. C /\ T e. D ) ) -> ( S ( X O Y ) T ) = (/) )
15	7, 14	sylanbr 490	. . 3 |- ( ( -. -. ( X e. A /\ Y e. B ) /\ -. ( S e. C /\ T e. D ) ) -> ( S ( X O Y ) T ) = (/) )
16	6, 15	jaoi3 1011	. 2 |- ( ( -. ( X e. A /\ Y e. B ) \/ -. ( S e. C /\ T e. D ) ) -> ( S ( X O Y ) T ) = (/) )
17	1, 16	sylbi 207	1 |- ( -. ( ( X e. A /\ Y e. B ) /\ ( S e. C /\ T e. D ) ) -> ( S ( X O Y ) T ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  wwlksnon0  26812