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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
Assertion
Ref Expression
enmap1lem1
Distinct variable groups:   ,,   ,
Allowed substitution hints:   ()   (,)

Proof of Theorem enmap1lem1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt 5652 . . 3
2 enmap1lem1.1 . . 3
3 opelres 4950 . . . . 5 Compose Compose
4 elima 4754 . . . . . . . . 9 Compose Compose
5 trtxp 5781 . . . . . . . . . . . 12
6 brco 4883 . . . . . . . . . . . . . 14
7 ancom 437 . . . . . . . . . . . . . . . 16
8 brin 4693 . . . . . . . . . . . . . . . . . 18
9 vex 2862 . . . . . . . . . . . . . . . . . . . . 21
10 brxp 4812 . . . . . . . . . . . . . . . . . . . . 21
119, 10mpbiran2 885 . . . . . . . . . . . . . . . . . . . 20
12 eliniseg 5020 . . . . . . . . . . . . . . . . . . . 20
1311, 12bitri 240 . . . . . . . . . . . . . . . . . . 19
1413anbi1i 676 . . . . . . . . . . . . . . . . . 18
15 vex 2862 . . . . . . . . . . . . . . . . . . 19
1615, 9op1st2nd 5790 . . . . . . . . . . . . . . . . . 18
178, 14, 163bitri 262 . . . . . . . . . . . . . . . . 17
1817anbi1i 676 . . . . . . . . . . . . . . . 16
197, 18bitri 240 . . . . . . . . . . . . . . 15
2019exbii 1582 . . . . . . . . . . . . . 14
2115, 9opex 4588 . . . . . . . . . . . . . . 15
22 breq2 4643 . . . . . . . . . . . . . . 15
2321, 22ceqsexv 2894 . . . . . . . . . . . . . 14
246, 20, 233bitri 262 . . . . . . . . . . . . 13
2524anbi1i 676 . . . . . . . . . . . 12
26 vex 2862 . . . . . . . . . . . . 13
2721, 26op1st2nd 5790 . . . . . . . . . . . 12
285, 25, 273bitri 262 . . . . . . . . . . 11
2928rexbii 2639 . . . . . . . . . 10 Compose Compose
30 risset 2661 . . . . . . . . . 10 Compose Compose
3129, 30bitr4i 243 . . . . . . . . 9 Compose Compose
324, 31bitri 240 . . . . . . . 8 Compose Compose
33 df-br 4640 . . . . . . . 8 Compose Compose
3432, 33bitr4i 243 . . . . . . 7 Compose Compose
35 brcomposeg 5819 . . . . . . . 8 Compose
3615, 9, 35mp2an 653 . . . . . . 7 Compose
37 eqcom 2355 . . . . . . 7
3834, 36, 373bitri 262 . . . . . 6 Compose
3938anbi2ci 677 . . . . 5 Compose
403, 39bitri 240 . . . 4 Compose
4140opabbi2i 4866 . . 3 Compose
421, 2, 413eqtr4i 2383 . 2 Compose
43 1stex 4739 . . . . . . . . . 10
4443cnvex 5102 . . . . . . . . 9
45 snex 4111 . . . . . . . . 9
4644, 45imaex 4747 . . . . . . . 8
47 vvex 4109 . . . . . . . 8
4846, 47xpex 5115 . . . . . . 7
49 2ndex 5112 . . . . . . 7
5048, 49inex 4105 . . . . . 6
5150, 43coex 4750 . . . . 5
5251, 49txpex 5785 . . . 4
53 composeex 5820 . . . 4 Compose
5452, 53imaex 4747 . . 3 Compose
55 ovex 5551 . . 3
5654, 55resex 5117 . 2 Compose
5742, 56eqeltri 2423 1
 Colors of variables: wff setvar class Syntax hints:   wb 176   wa 358  wex 1541   wceq 1642   wcel 1710  wrex 2615  cvv 2859   cin 3208  csn 3737  cop 4561  copab 4622   class class class wbr 4639  c1st 4717   ccom 4721  cima 4722   cxp 4770  ccnv 4771   cres 4774  c2nd 4783  (class class class)co 5525   cmpt 5651   ctxp 5735   Compose ccompose 5747   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-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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