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Theorem enprmaplem1 6076
 Description: Lemma for enprmap 6082. Set up stratification. (Contributed by SF, 3-Mar-2015.)
Hypothesis
Ref Expression
enprmaplem1.1 W = (r (Am B) (r “ {x}))
Assertion
Ref Expression
enprmaplem1 W V
Distinct variable groups:   A,r   B,r   x,r
Allowed substitution hints:   A(x)   B(x)   W(x,r)

Proof of Theorem enprmaplem1
Dummy variables y t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enprmaplem1.1 . . 3 W = (r (Am B) (r “ {x}))
2 elima1c 4947 . . . . . . 7 ({y}, r (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c) ↔ t{t}, {y}, r ( SI (1st (2nd “ {x})) ⊗ S ))
3 oteltxp 5782 . . . . . . . . 9 ({t}, {y}, r ( SI (1st (2nd “ {x})) ⊗ S ) ↔ ({t}, {y} SI (1st (2nd “ {x})) {t}, r S ))
4 vex 2862 . . . . . . . . . . . 12 t V
5 vex 2862 . . . . . . . . . . . 12 y V
64, 5opsnelsi 5774 . . . . . . . . . . 11 ({t}, {y} SI (1st (2nd “ {x})) ↔ t, y (1st (2nd “ {x})))
7 df-br 4640 . . . . . . . . . . . 12 (t(1st (2nd “ {x}))yt, y (1st (2nd “ {x})))
8 brres 4949 . . . . . . . . . . . . 13 (t(1st (2nd “ {x}))y ↔ (t1st y t (2nd “ {x})))
9 eliniseg 5020 . . . . . . . . . . . . . 14 (t (2nd “ {x}) ↔ t2nd x)
109anbi2i 675 . . . . . . . . . . . . 13 ((t1st y t (2nd “ {x})) ↔ (t1st y t2nd x))
118, 10bitri 240 . . . . . . . . . . . 12 (t(1st (2nd “ {x}))y ↔ (t1st y t2nd x))
127, 11bitr3i 242 . . . . . . . . . . 11 (t, y (1st (2nd “ {x})) ↔ (t1st y t2nd x))
13 vex 2862 . . . . . . . . . . . 12 x V
145, 13op1st2nd 5790 . . . . . . . . . . 11 ((t1st y t2nd x) ↔ t = y, x)
156, 12, 143bitri 262 . . . . . . . . . 10 ({t}, {y} SI (1st (2nd “ {x})) ↔ t = y, x)
16 vex 2862 . . . . . . . . . . 11 r V
174, 16opelssetsn 4760 . . . . . . . . . 10 ({t}, r S t r)
1815, 17anbi12i 678 . . . . . . . . 9 (({t}, {y} SI (1st (2nd “ {x})) {t}, r S ) ↔ (t = y, x t r))
193, 18bitri 240 . . . . . . . 8 ({t}, {y}, r ( SI (1st (2nd “ {x})) ⊗ S ) ↔ (t = y, x t r))
2019exbii 1582 . . . . . . 7 (t{t}, {y}, r ( SI (1st (2nd “ {x})) ⊗ S ) ↔ t(t = y, x t r))
212, 20bitri 240 . . . . . 6 ({y}, r (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c) ↔ t(t = y, x t r))
225, 13opex 4588 . . . . . . . 8 y, x V
23 eleq1 2413 . . . . . . . 8 (t = y, x → (t ry, x r))
2422, 23ceqsexv 2894 . . . . . . 7 (t(t = y, x t r) ↔ y, x r)
25 df-br 4640 . . . . . . 7 (yrxy, x r)
2624, 25bitr4i 243 . . . . . 6 (t(t = y, x t r) ↔ yrx)
2721, 26bitri 240 . . . . 5 ({y}, r (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c) ↔ yrx)
28 eliniseg 5020 . . . . 5 (y (r “ {x}) ↔ yrx)
2927, 28bitr4i 243 . . . 4 ({y}, r (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c) ↔ y (r “ {x}))
3029releqmpt 5808 . . 3 (((Am B) × V) ∩ ∼ (( Ins3 S Ins2 (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c)) “ 1c)) = (r (Am B) (r “ {x}))
311, 30eqtr4i 2376 . 2 W = (((Am B) × V) ∩ ∼ (( Ins3 S Ins2 (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c)) “ 1c))
32 ovex 5551 . . 3 (Am B) V
33 1stex 4739 . . . . . . 7 1st V
34 2ndex 5112 . . . . . . . . 9 2nd V
3534cnvex 5102 . . . . . . . 8 2nd V
36 snex 4111 . . . . . . . 8 {x} V
3735, 36imaex 4747 . . . . . . 7 (2nd “ {x}) V
3833, 37resex 5117 . . . . . 6 (1st (2nd “ {x})) V
3938siex 4753 . . . . 5 SI (1st (2nd “ {x})) V
40 ssetex 4744 . . . . 5 S V
4139, 40txpex 5785 . . . 4 ( SI (1st (2nd “ {x})) ⊗ S ) V
42 1cex 4142 . . . 4 1c V
4341, 42imaex 4747 . . 3 (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c) V
4432, 43mptexlem 5810 . 2 (((Am B) × V) ∩ ∼ (( Ins3 S Ins2 (( SI (1st (2nd “ {x})) ⊗ S ) “ 1c)) “ 1c)) V
4531, 44eqeltri 2423 1 W V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208   ⊕ csymdif 3209  {csn 3737  1cc1c 4134  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   S csset 4719   SI csi 4720   “ cima 4722   × cxp 4770  ◡ccnv 4771   ↾ cres 4774  2nd c2nd 4783  (class class class)co 5525   ↦ cmpt 5651   ⊗ ctxp 5735   Ins2 cins2 5749   Ins3 cins3 5751   ↑m cmap 5999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fv 4795  df-2nd 4797  df-ov 5526  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752 This theorem is referenced by:  enprmap  6082
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