Theorem fmucndlem 22095

Description: Lemma for fmucnd 22096. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Assertion

Ref	Expression
fmucndlem	|- ( ( F Fn X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( A X. A ) ) = ( ( F " A ) X. ( F " A ) ) )

Distinct variable groups:   x, y, A   x, F, y   x, X, y

Proof of Theorem fmucndlem

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5127	. . 3 |- ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( A X. A ) ) = ran ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) )
2		simpr 477	. . . . 5 |- ( ( F Fn X /\ A C_ X ) -> A C_ X )
3		resmpt2 6758	. . . . 5 |- ( ( A C_ X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) )
4	2, 3	sylancom 701	. . . 4 |- ( ( F Fn X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) )
5	4	rneqd 5353	. . 3 |- ( ( F Fn X /\ A C_ X ) -> ran ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) ) = ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) )
6	1, 5	syl5eq 2668	. 2 |- ( ( F Fn X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( A X. A ) ) = ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) )
7		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
8		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
9	7, 8	op1std 7178	. . . . . . . . . . . 12 |- ( p = <. x , y >. -> ( 1st ` p ) = x )
10	9	fveq2d 6195	. . . . . . . . . . 11 |- ( p = <. x , y >. -> ( F ` ( 1st ` p ) ) = ( F ` x ) )
11	7, 8	op2ndd 7179	. . . . . . . . . . . 12 |- ( p = <. x , y >. -> ( 2nd ` p ) = y )
12	11	fveq2d 6195	. . . . . . . . . . 11 |- ( p = <. x , y >. -> ( F ` ( 2nd ` p ) ) = ( F ` y ) )
13	10, 12	opeq12d 4410	. . . . . . . . . 10 |- ( p = <. x , y >. -> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. = <. ( F ` x ) , ( F ` y ) >. )
14	13	mpt2mpt 6752	. . . . . . . . 9 |- ( p e. ( A X. A ) |-> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. )
15	14	eqcomi 2631	. . . . . . . 8 |- ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ( p e. ( A X. A ) |-> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. )
16	15	rneqi 5352	. . . . . . 7 |- ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ran ( p e. ( A X. A ) |-> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. )
17		fvexd 6203	. . . . . . 7 |- ( ( T. /\ p e. ( A X. A ) ) -> ( F ` ( 1st ` p ) ) e. _V )
18		fvexd 6203	. . . . . . 7 |- ( ( T. /\ p e. ( A X. A ) ) -> ( F ` ( 2nd ` p ) ) e. _V )
19	16, 17, 18	fliftrel 6558	. . . . . 6 |- ( T. -> ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) C_ ( _V X. _V ) )
20	19	trud 1493	. . . . 5 |- ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) C_ ( _V X. _V )
21	20	sseli 3599	. . . 4 |- ( p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) -> p e. ( _V X. _V ) )
22	21	adantl 482	. . 3 |- ( ( ( F Fn X /\ A C_ X ) /\ p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) -> p e. ( _V X. _V ) )
23		xpss 5226	. . . . 5 |- ( ( F " A ) X. ( F " A ) ) C_ ( _V X. _V )
24	23	sseli 3599	. . . 4 |- ( p e. ( ( F " A ) X. ( F " A ) ) -> p e. ( _V X. _V ) )
25	24	adantl 482	. . 3 |- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( ( F " A ) X. ( F " A ) ) ) -> p e. ( _V X. _V ) )
26		fvelimab 6253	. . . . . . . 8 |- ( ( F Fn X /\ A C_ X ) -> ( ( 1st ` p ) e. ( F " A ) <-> E. x e. A ( F ` x ) = ( 1st ` p ) ) )
27		fvelimab 6253	. . . . . . . 8 |- ( ( F Fn X /\ A C_ X ) -> ( ( 2nd ` p ) e. ( F " A ) <-> E. y e. A ( F ` y ) = ( 2nd ` p ) ) )
28	26, 27	anbi12d 747	. . . . . . 7 |- ( ( F Fn X /\ A C_ X ) -> ( ( ( 1st ` p ) e. ( F " A ) /\ ( 2nd ` p ) e. ( F " A ) ) <-> ( E. x e. A ( F ` x ) = ( 1st ` p ) /\ E. y e. A ( F ` y ) = ( 2nd ` p ) ) ) )
29		eqid 2622	. . . . . . . . 9 |- ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. )
30		opex 4932	. . . . . . . . 9 |- <. ( F ` x ) , ( F ` y ) >. e. _V
31	29, 30	elrnmpt2 6773	. . . . . . . 8 |- ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> E. x e. A E. y e. A <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. )
32		eqcom 2629	. . . . . . . . . 10 |- ( <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. <-> <. ( F ` x ) , ( F ` y ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. )
33		fvex 6201	. . . . . . . . . . 11 |- ( 1st ` p ) e. _V
34		fvex 6201	. . . . . . . . . . 11 |- ( 2nd ` p ) e. _V
35	33, 34	opth2 4949	. . . . . . . . . 10 |- ( <. ( F ` x ) , ( F ` y ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. <-> ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) )
36	32, 35	bitri 264	. . . . . . . . 9 |- ( <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. <-> ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) )
37	36	2rexbii 3042	. . . . . . . 8 |- ( E. x e. A E. y e. A <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. <-> E. x e. A E. y e. A ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) )
38		reeanv 3107	. . . . . . . 8 |- ( E. x e. A E. y e. A ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) <-> ( E. x e. A ( F ` x ) = ( 1st ` p ) /\ E. y e. A ( F ` y ) = ( 2nd ` p ) ) )
39	31, 37, 38	3bitri 286	. . . . . . 7 |- ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> ( E. x e. A ( F ` x ) = ( 1st ` p ) /\ E. y e. A ( F ` y ) = ( 2nd ` p ) ) )
40	28, 39	syl6rbbr 279	. . . . . 6 |- ( ( F Fn X /\ A C_ X ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> ( ( 1st ` p ) e. ( F " A ) /\ ( 2nd ` p ) e. ( F " A ) ) ) )
41		opelxp 5146	. . . . . 6 |- ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) <-> ( ( 1st ` p ) e. ( F " A ) /\ ( 2nd ` p ) e. ( F " A ) ) )
42	40, 41	syl6bbr 278	. . . . 5 |- ( ( F Fn X /\ A C_ X ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) ) )
43	42	adantr 481	. . . 4 |- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) ) )
44		1st2nd2 7205	. . . . . 6 |- ( p e. ( _V X. _V ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
45	44	adantl 482	. . . . 5 |- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
46	45	eleq1d 2686	. . . 4 |- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) )
47	45	eleq1d 2686	. . . 4 |- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( p e. ( ( F " A ) X. ( F " A ) ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) ) )
48	43, 46, 47	3bitr4d 300	. . 3 |- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> p e. ( ( F " A ) X. ( F " A ) ) ) )
49	22, 25, 48	eqrdav 2621	. 2 |- ( ( F Fn X /\ A C_ X ) -> ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ( ( F " A ) X. ( F " A ) ) )
50	6, 49	eqtrd 2656	1 |- ( ( F Fn X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( A X. A ) ) = ( ( F " A ) X. ( F " A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   ` cfv 5888   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

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  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  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-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-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  fmucnd  22096