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Theorem enmap1lem1 6069
 Description: Lemma for enmap1 6074. Set up stratification. (Contributed by SF, 3-Mar-2015.)
Hypothesis
Ref Expression
enmap1lem1.1 W = (s (Am G) (r s))
Assertion
Ref Expression
enmap1lem1 W V
Distinct variable groups:   A,r,s   G,s
Allowed substitution hints:   G(r)   W(s,r)

Proof of Theorem enmap1lem1
Dummy variables p x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt 5652 . . 3 (s (Am G) (r s)) = {s, x (s (Am G) x = (r s))}
2 enmap1lem1.1 . . 3 W = (s (Am G) (r s))
3 opelres 4950 . . . . 5 (s, x (((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) (Am G)) ↔ (s, x ((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) s (Am G)))
4 elima 4754 . . . . . . . . 9 (s, x ((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) ↔ p Compose p(((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd )s, x)
5 trtxp 5781 . . . . . . . . . . . 12 (p(((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd )s, x ↔ (p((((1st “ {r}) × V) ∩ 2nd ) 1st )s p2nd x))
6 brco 4883 . . . . . . . . . . . . . 14 (p((((1st “ {r}) × V) ∩ 2nd ) 1st )sx(p1st x x(((1st “ {r}) × V) ∩ 2nd )s))
7 ancom 437 . . . . . . . . . . . . . . . 16 ((p1st x x(((1st “ {r}) × V) ∩ 2nd )s) ↔ (x(((1st “ {r}) × V) ∩ 2nd )s p1st x))
8 brin 4693 . . . . . . . . . . . . . . . . . 18 (x(((1st “ {r}) × V) ∩ 2nd )s ↔ (x((1st “ {r}) × V)s x2nd s))
9 vex 2862 . . . . . . . . . . . . . . . . . . . . 21 s V
10 brxp 4812 . . . . . . . . . . . . . . . . . . . . 21 (x((1st “ {r}) × V)s ↔ (x (1st “ {r}) s V))
119, 10mpbiran2 885 . . . . . . . . . . . . . . . . . . . 20 (x((1st “ {r}) × V)sx (1st “ {r}))
12 eliniseg 5020 . . . . . . . . . . . . . . . . . . . 20 (x (1st “ {r}) ↔ x1st r)
1311, 12bitri 240 . . . . . . . . . . . . . . . . . . 19 (x((1st “ {r}) × V)sx1st r)
1413anbi1i 676 . . . . . . . . . . . . . . . . . 18 ((x((1st “ {r}) × V)s x2nd s) ↔ (x1st r x2nd s))
15 vex 2862 . . . . . . . . . . . . . . . . . . 19 r V
1615, 9op1st2nd 5790 . . . . . . . . . . . . . . . . . 18 ((x1st r x2nd s) ↔ x = r, s)
178, 14, 163bitri 262 . . . . . . . . . . . . . . . . 17 (x(((1st “ {r}) × V) ∩ 2nd )sx = r, s)
1817anbi1i 676 . . . . . . . . . . . . . . . 16 ((x(((1st “ {r}) × V) ∩ 2nd )s p1st x) ↔ (x = r, s p1st x))
197, 18bitri 240 . . . . . . . . . . . . . . 15 ((p1st x x(((1st “ {r}) × V) ∩ 2nd )s) ↔ (x = r, s p1st x))
2019exbii 1582 . . . . . . . . . . . . . 14 (x(p1st x x(((1st “ {r}) × V) ∩ 2nd )s) ↔ x(x = r, s p1st x))
2115, 9opex 4588 . . . . . . . . . . . . . . 15 r, s V
22 breq2 4643 . . . . . . . . . . . . . . 15 (x = r, s → (p1st xp1st r, s))
2321, 22ceqsexv 2894 . . . . . . . . . . . . . 14 (x(x = r, s p1st x) ↔ p1st r, s)
246, 20, 233bitri 262 . . . . . . . . . . . . 13 (p((((1st “ {r}) × V) ∩ 2nd ) 1st )sp1st r, s)
2524anbi1i 676 . . . . . . . . . . . 12 ((p((((1st “ {r}) × V) ∩ 2nd ) 1st )s p2nd x) ↔ (p1st r, s p2nd x))
26 vex 2862 . . . . . . . . . . . . 13 x V
2721, 26op1st2nd 5790 . . . . . . . . . . . 12 ((p1st r, s p2nd x) ↔ p = r, s, x)
285, 25, 273bitri 262 . . . . . . . . . . 11 (p(((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd )s, xp = r, s, x)
2928rexbii 2639 . . . . . . . . . 10 (p Compose p(((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd )s, xp Compose p = r, s, x)
30 risset 2661 . . . . . . . . . 10 (r, s, x Composep Compose p = r, s, x)
3129, 30bitr4i 243 . . . . . . . . 9 (p Compose p(((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd )s, xr, s, x Compose )
324, 31bitri 240 . . . . . . . 8 (s, x ((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) ↔ r, s, x Compose )
33 df-br 4640 . . . . . . . 8 (r, s Compose xr, s, x Compose )
3432, 33bitr4i 243 . . . . . . 7 (s, x ((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) ↔ r, s Compose x)
35 brcomposeg 5819 . . . . . . . 8 ((r V s V) → (r, s Compose x ↔ (r s) = x))
3615, 9, 35mp2an 653 . . . . . . 7 (r, s Compose x ↔ (r s) = x)
37 eqcom 2355 . . . . . . 7 ((r s) = xx = (r s))
3834, 36, 373bitri 262 . . . . . 6 (s, x ((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) ↔ x = (r s))
3938anbi2ci 677 . . . . 5 ((s, x ((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) s (Am G)) ↔ (s (Am G) x = (r s)))
403, 39bitri 240 . . . 4 (s, x (((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) (Am G)) ↔ (s (Am G) x = (r s)))
4140opabbi2i 4866 . . 3 (((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) (Am G)) = {s, x (s (Am G) x = (r s))}
421, 2, 413eqtr4i 2383 . 2 W = (((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) (Am G))
43 1stex 4739 . . . . . . . . . 10 1st V
4443cnvex 5102 . . . . . . . . 9 1st V
45 snex 4111 . . . . . . . . 9 {r} V
4644, 45imaex 4747 . . . . . . . 8 (1st “ {r}) V
47 vvex 4109 . . . . . . . 8 V V
4846, 47xpex 5115 . . . . . . 7 ((1st “ {r}) × V) V
49 2ndex 5112 . . . . . . 7 2nd V
5048, 49inex 4105 . . . . . 6 (((1st “ {r}) × V) ∩ 2nd ) V
5150, 43coex 4750 . . . . 5 ((((1st “ {r}) × V) ∩ 2nd ) 1st ) V
5251, 49txpex 5785 . . . 4 (((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) V
53 composeex 5820 . . . 4 Compose V
5452, 53imaex 4747 . . 3 ((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) V
55 ovex 5551 . . 3 (Am G) V
5654, 55resex 5117 . 2 (((((((1st “ {r}) × V) ∩ 2nd ) 1st ) ⊗ 2nd ) “ Compose ) (Am G)) 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:  enmap1  6074
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