Theorem el2mpt2cl 7251

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. Using implicit substitution. (Contributed by AV, 21-May-2021.)

Hypotheses

Ref	Expression
el2mpt2cl.o	|- O = ( x e. A , y e. B |-> ( s e. C , t e. D |-> E ) )
el2mpt2cl.e	|- ( ( x = X /\ y = Y ) -> ( C = F /\ D = G ) )

Assertion

Ref	Expression
el2mpt2cl	|- ( A. x e. A A. y e. B ( C e. U /\ D e. V ) -> ( W e. ( S ( X O Y ) T ) -> ( ( X e. A /\ Y e. B ) /\ ( S e. F /\ T e. G ) ) ) )

Distinct variable groups:   A, s, t, x, y   B, s, t, x, y   C, s, t   D, s, t   x, F, y   x, G, y   x, U, y   x, V, y   X, s, t, x, y   Y, s, t, x, y

Allowed substitution hints:   C( x, y)   D( x, y)   S( x, y, t, s)   T( x, y, t, s)   U( t, s)   E( x, y, t, s)   F( t, s)   G( t, s)   O( x, y, t, s)   V( t, s)   W( x, y, t, s)

Proof of Theorem el2mpt2cl

Step	Hyp	Ref	Expression
1		el2mpt2cl.o	. . 3 |- O = ( x e. A , y e. B |-> ( s e. C , t e. D |-> E ) )
2	1	el2mpt2csbcl 7250	. 2 |- ( A. x e. A A. y e. B ( C e. U /\ D e. V ) -> ( W e. ( S ( X O Y ) T ) -> ( ( X e. A /\ Y e. B ) /\ ( S e. [_ X / x ]_ [_ Y / y ]_ C /\ T e. [_ X / x ]_ [_ Y / y ]_ D ) ) ) )
3		simpl 473	. . . . . . 7 |- ( ( X e. A /\ Y e. B ) -> X e. A )
4		simplr 792	. . . . . . . 8 |- ( ( ( X e. A /\ Y e. B ) /\ x = X ) -> Y e. B )
5		el2mpt2cl.e	. . . . . . . . . 10 |- ( ( x = X /\ y = Y ) -> ( C = F /\ D = G ) )
6	5	simpld 475	. . . . . . . . 9 |- ( ( x = X /\ y = Y ) -> C = F )
7	6	adantll 750	. . . . . . . 8 |- ( ( ( ( X e. A /\ Y e. B ) /\ x = X ) /\ y = Y ) -> C = F )
8	4, 7	csbied 3560	. . . . . . 7 |- ( ( ( X e. A /\ Y e. B ) /\ x = X ) -> [_ Y / y ]_ C = F )
9	3, 8	csbied 3560	. . . . . 6 |- ( ( X e. A /\ Y e. B ) -> [_ X / x ]_ [_ Y / y ]_ C = F )
10	9	eleq2d 2687	. . . . 5 |- ( ( X e. A /\ Y e. B ) -> ( S e. [_ X / x ]_ [_ Y / y ]_ C <-> S e. F ) )
11	5	simprd 479	. . . . . . . . 9 |- ( ( x = X /\ y = Y ) -> D = G )
12	11	adantll 750	. . . . . . . 8 |- ( ( ( ( X e. A /\ Y e. B ) /\ x = X ) /\ y = Y ) -> D = G )
13	4, 12	csbied 3560	. . . . . . 7 |- ( ( ( X e. A /\ Y e. B ) /\ x = X ) -> [_ Y / y ]_ D = G )
14	3, 13	csbied 3560	. . . . . 6 |- ( ( X e. A /\ Y e. B ) -> [_ X / x ]_ [_ Y / y ]_ D = G )
15	14	eleq2d 2687	. . . . 5 |- ( ( X e. A /\ Y e. B ) -> ( T e. [_ X / x ]_ [_ Y / y ]_ D <-> T e. G ) )
16	10, 15	anbi12d 747	. . . 4 |- ( ( X e. A /\ Y e. B ) -> ( ( S e. [_ X / x ]_ [_ Y / y ]_ C /\ T e. [_ X / x ]_ [_ Y / y ]_ D ) <-> ( S e. F /\ T e. G ) ) )
17	16	biimpd 219	. . 3 |- ( ( X e. A /\ Y e. B ) -> ( ( S e. [_ X / x ]_ [_ Y / y ]_ C /\ T e. [_ X / x ]_ [_ Y / y ]_ D ) -> ( S e. F /\ T e. G ) ) )
18	17	imdistani 726	. 2 |- ( ( ( X e. A /\ Y e. B ) /\ ( S e. [_ X / x ]_ [_ Y / y ]_ C /\ T e. [_ X / x ]_ [_ Y / y ]_ D ) ) -> ( ( X e. A /\ Y e. B ) /\ ( S e. F /\ T e. G ) ) )
19	2, 18	syl6 35	1 |- ( A. x e. A A. y e. B ( C e. U /\ D e. V ) -> ( W e. ( S ( X O Y ) T ) -> ( ( X e. A /\ Y e. B ) /\ ( S e. F /\ T e. G ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   [_csb 3533  (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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  wspthnonp  26744