New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  enmap2lem1 GIF version

Theorem enmap2lem1 6063
 Description: Lemma for enmap2 6068. Set up stratification. (Contributed by SF, 26-Feb-2015.)
Hypothesis
Ref Expression
enmap2lem1.1 W = (s (Gm A) (s r))
Assertion
Ref Expression
enmap2lem1 W V
Distinct variable groups:   A,s   G,s   s,r
Allowed substitution hints:   A(r)   G(r)   W(s,r)

Proof of Theorem enmap2lem1
Dummy variables p x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt 5652 . . 3 (s (Gm A) (s r)) = {s, x (s (Gm A) x = (s r))}
2 enmap2lem1.1 . . 3 W = (s (Gm A) (s r))
3 opelres 4950 . . . . 5 (s, x (((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) (Gm A)) ↔ (s, x ((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) s (Gm A)))
4 trtxp 5781 . . . . . . . . . 10 (p(((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd )s, x ↔ (p((1st ∩ ((2nd “ {r}) × V)) 1st )s p2nd x))
5 brco 4883 . . . . . . . . . . . 12 (p((1st ∩ ((2nd “ {r}) × V)) 1st )sx(p1st x x(1st ∩ ((2nd “ {r}) × V))s))
6 ancom 437 . . . . . . . . . . . . . 14 ((p1st x x(1st ∩ ((2nd “ {r}) × V))s) ↔ (x(1st ∩ ((2nd “ {r}) × V))s p1st x))
7 brin 4693 . . . . . . . . . . . . . . . 16 (x(1st ∩ ((2nd “ {r}) × V))s ↔ (x1st s x((2nd “ {r}) × V)s))
8 vex 2862 . . . . . . . . . . . . . . . . . . 19 s V
9 brxp 4812 . . . . . . . . . . . . . . . . . . 19 (x((2nd “ {r}) × V)s ↔ (x (2nd “ {r}) s V))
108, 9mpbiran2 885 . . . . . . . . . . . . . . . . . 18 (x((2nd “ {r}) × V)sx (2nd “ {r}))
11 eliniseg 5020 . . . . . . . . . . . . . . . . . 18 (x (2nd “ {r}) ↔ x2nd r)
1210, 11bitri 240 . . . . . . . . . . . . . . . . 17 (x((2nd “ {r}) × V)sx2nd r)
1312anbi2i 675 . . . . . . . . . . . . . . . 16 ((x1st s x((2nd “ {r}) × V)s) ↔ (x1st s x2nd r))
14 vex 2862 . . . . . . . . . . . . . . . . . 18 r V
1514cnvex 5102 . . . . . . . . . . . . . . . . 17 r V
168, 15op1st2nd 5790 . . . . . . . . . . . . . . . 16 ((x1st s x2nd r) ↔ x = s, r)
177, 13, 163bitri 262 . . . . . . . . . . . . . . 15 (x(1st ∩ ((2nd “ {r}) × V))sx = s, r)
1817anbi1i 676 . . . . . . . . . . . . . 14 ((x(1st ∩ ((2nd “ {r}) × V))s p1st x) ↔ (x = s, r p1st x))
196, 18bitri 240 . . . . . . . . . . . . 13 ((p1st x x(1st ∩ ((2nd “ {r}) × V))s) ↔ (x = s, r p1st x))
2019exbii 1582 . . . . . . . . . . . 12 (x(p1st x x(1st ∩ ((2nd “ {r}) × V))s) ↔ x(x = s, r p1st x))
218, 15opex 4588 . . . . . . . . . . . . 13 s, r V
22 breq2 4643 . . . . . . . . . . . . 13 (x = s, r → (p1st xp1st s, r))
2321, 22ceqsexv 2894 . . . . . . . . . . . 12 (x(x = s, r p1st x) ↔ p1st s, r)
245, 20, 233bitri 262 . . . . . . . . . . 11 (p((1st ∩ ((2nd “ {r}) × V)) 1st )sp1st s, r)
2524anbi1i 676 . . . . . . . . . 10 ((p((1st ∩ ((2nd “ {r}) × V)) 1st )s p2nd x) ↔ (p1st s, r p2nd x))
26 vex 2862 . . . . . . . . . . 11 x V
2721, 26op1st2nd 5790 . . . . . . . . . 10 ((p1st s, r p2nd x) ↔ p = s, r, x)
284, 25, 273bitri 262 . . . . . . . . 9 (p(((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd )s, xp = s, r, x)
2928rexbii 2639 . . . . . . . 8 (p Compose p(((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd )s, xp Compose p = s, r, x)
30 elima 4754 . . . . . . . 8 (s, x ((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) ↔ p Compose p(((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd )s, x)
31 risset 2661 . . . . . . . 8 (s, r, x Composep Compose p = s, r, x)
3229, 30, 313bitr4i 268 . . . . . . 7 (s, x ((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) ↔ s, r, x Compose )
33 df-br 4640 . . . . . . 7 (s, r Compose xs, r, x Compose )
34 brcomposeg 5819 . . . . . . . . 9 ((s V r V) → (s, r Compose x ↔ (s r) = x))
358, 15, 34mp2an 653 . . . . . . . 8 (s, r Compose x ↔ (s r) = x)
36 eqcom 2355 . . . . . . . 8 ((s r) = xx = (s r))
3735, 36bitri 240 . . . . . . 7 (s, r Compose xx = (s r))
3832, 33, 373bitr2i 264 . . . . . 6 (s, x ((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) ↔ x = (s r))
3938anbi2ci 677 . . . . 5 ((s, x ((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) s (Gm A)) ↔ (s (Gm A) x = (s r)))
403, 39bitri 240 . . . 4 (s, x (((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) (Gm A)) ↔ (s (Gm A) x = (s r)))
4140opabbi2i 4866 . . 3 (((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) (Gm A)) = {s, x (s (Gm A) x = (s r))}
421, 2, 413eqtr4i 2383 . 2 W = (((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) (Gm A))
43 1stex 4739 . . . . . . 7 1st V
44 2ndex 5112 . . . . . . . . . 10 2nd V
4544cnvex 5102 . . . . . . . . 9 2nd V
46 snex 4111 . . . . . . . . 9 {r} V
4745, 46imaex 4747 . . . . . . . 8 (2nd “ {r}) V
48 vvex 4109 . . . . . . . 8 V V
4947, 48xpex 5115 . . . . . . 7 ((2nd “ {r}) × V) V
5043, 49inex 4105 . . . . . 6 (1st ∩ ((2nd “ {r}) × V)) V
5150, 43coex 4750 . . . . 5 ((1st ∩ ((2nd “ {r}) × V)) 1st ) V
5251, 44txpex 5785 . . . 4 (((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) V
53 composeex 5820 . . . 4 Compose V
5452, 53imaex 4747 . . 3 ((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) V
55 ovex 5551 . . 3 (Gm A) V
5654, 55resex 5117 . 2 (((((1st ∩ ((2nd “ {r}) × V)) 1st ) ⊗ 2nd ) “ Compose ) (Gm A)) V
5742, 56eqeltri 2423 1 W V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∩ cin 3208  {csn 3737  ⟨cop 4561  {copab 4622   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721   “ cima 4722   × cxp 4770  ◡ccnv 4771   ↾ cres 4774  2nd c2nd 4783  (class class class)co 5525   ↦ cmpt 5651   ⊗ ctxp 5735   Compose ccompose 5747   ↑m cmap 5999 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758 This theorem is referenced by:  enmap2  6068
 Copyright terms: Public domain W3C validator