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

Theorem txpcofun 5803
 Description: Composition distributes over tail cross product in the case of a function. (Contributed by SF, 18-Feb-2015.)
Hypothesis
Ref Expression
txpcofun.1 Fun F
Assertion
Ref Expression
txpcofun ((RS) F) = ((R F) ⊗ (S F))

Proof of Theorem txpcofun
Dummy variables t x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . 4 t V
2 opeqex 4621 . . . 4 (t V → yz t = y, z)
31, 2ax-mp 5 . . 3 yz t = y, z
4 dmcoss 4971 . . . . . . . . . 10 dom (R F) dom F
5 opeldm 4910 . . . . . . . . . 10 (x, y (R F) → x dom (R F))
64, 5sseldi 3271 . . . . . . . . 9 (x, y (R F) → x dom F)
76pm4.71ri 614 . . . . . . . 8 (x, y (R F) ↔ (x dom F x, y (R F)))
87anbi1i 676 . . . . . . 7 ((x, y (R F) x, z (S F)) ↔ ((x dom F x, y (R F)) x, z (S F)))
9 anass 630 . . . . . . 7 (((x dom F x, y (R F)) x, z (S F)) ↔ (x dom F (x, y (R F) x, z (S F))))
10 fvex 5339 . . . . . . . . . . 11 (Fx) V
11 breq1 4642 . . . . . . . . . . 11 (t = (Fx) → (tRy ↔ (Fx)Ry))
1210, 11ceqsexv 2894 . . . . . . . . . 10 (t(t = (Fx) tRy) ↔ (Fx)Ry)
13 breq1 4642 . . . . . . . . . . 11 (t = (Fx) → (tSz ↔ (Fx)Sz))
1410, 13ceqsexv 2894 . . . . . . . . . 10 (t(t = (Fx) tSz) ↔ (Fx)Sz)
1512, 14anbi12i 678 . . . . . . . . 9 ((t(t = (Fx) tRy) t(t = (Fx) tSz)) ↔ ((Fx)Ry (Fx)Sz))
16 eqcom 2355 . . . . . . . . . . . . . 14 (t = (Fx) ↔ (Fx) = t)
17 txpcofun.1 . . . . . . . . . . . . . . 15 Fun F
18 funbrfvb 5360 . . . . . . . . . . . . . . 15 ((Fun F x dom F) → ((Fx) = txFt))
1917, 18mpan 651 . . . . . . . . . . . . . 14 (x dom F → ((Fx) = txFt))
2016, 19syl5bb 248 . . . . . . . . . . . . 13 (x dom F → (t = (Fx) ↔ xFt))
2120anbi1d 685 . . . . . . . . . . . 12 (x dom F → ((t = (Fx) tRy) ↔ (xFt tRy)))
2221exbidv 1626 . . . . . . . . . . 11 (x dom F → (t(t = (Fx) tRy) ↔ t(xFt tRy)))
23 opelco 4884 . . . . . . . . . . 11 (x, y (R F) ↔ t(xFt tRy))
2422, 23syl6bbr 254 . . . . . . . . . 10 (x dom F → (t(t = (Fx) tRy) ↔ x, y (R F)))
2520anbi1d 685 . . . . . . . . . . . 12 (x dom F → ((t = (Fx) tSz) ↔ (xFt tSz)))
2625exbidv 1626 . . . . . . . . . . 11 (x dom F → (t(t = (Fx) tSz) ↔ t(xFt tSz)))
27 opelco 4884 . . . . . . . . . . 11 (x, z (S F) ↔ t(xFt tSz))
2826, 27syl6bbr 254 . . . . . . . . . 10 (x dom F → (t(t = (Fx) tSz) ↔ x, z (S F)))
2924, 28anbi12d 691 . . . . . . . . 9 (x dom F → ((t(t = (Fx) tRy) t(t = (Fx) tSz)) ↔ (x, y (R F) x, z (S F))))
3015, 29syl5rbbr 251 . . . . . . . 8 (x dom F → ((x, y (R F) x, z (S F)) ↔ ((Fx)Ry (Fx)Sz)))
3130pm5.32i 618 . . . . . . 7 ((x dom F (x, y (R F) x, z (S F))) ↔ (x dom F ((Fx)Ry (Fx)Sz)))
328, 9, 313bitrri 263 . . . . . 6 ((x dom F ((Fx)Ry (Fx)Sz)) ↔ (x, y (R F) x, z (S F)))
33 opelco 4884 . . . . . . 7 (x, y, z ((RS) F) ↔ t(xFt t(RS)y, z))
34 19.41v 1901 . . . . . . . 8 (t(xFt ((Fx)Ry (Fx)Sz)) ↔ (t xFt ((Fx)Ry (Fx)Sz)))
35 funbrfv 5356 . . . . . . . . . . . 12 (Fun F → (xFt → (Fx) = t))
3617, 35ax-mp 5 . . . . . . . . . . 11 (xFt → (Fx) = t)
37 trtxp 5781 . . . . . . . . . . . 12 ((Fx)(RS)y, z ↔ ((Fx)Ry (Fx)Sz))
38 breq1 4642 . . . . . . . . . . . 12 ((Fx) = t → ((Fx)(RS)y, zt(RS)y, z))
3937, 38syl5rbbr 251 . . . . . . . . . . 11 ((Fx) = t → (t(RS)y, z ↔ ((Fx)Ry (Fx)Sz)))
4036, 39syl 15 . . . . . . . . . 10 (xFt → (t(RS)y, z ↔ ((Fx)Ry (Fx)Sz)))
4140pm5.32i 618 . . . . . . . . 9 ((xFt t(RS)y, z) ↔ (xFt ((Fx)Ry (Fx)Sz)))
4241exbii 1582 . . . . . . . 8 (t(xFt t(RS)y, z) ↔ t(xFt ((Fx)Ry (Fx)Sz)))
43 eldm 4898 . . . . . . . . 9 (x dom Ft xFt)
4443anbi1i 676 . . . . . . . 8 ((x dom F ((Fx)Ry (Fx)Sz)) ↔ (t xFt ((Fx)Ry (Fx)Sz)))
4534, 42, 443bitr4i 268 . . . . . . 7 (t(xFt t(RS)y, z) ↔ (x dom F ((Fx)Ry (Fx)Sz)))
4633, 45bitri 240 . . . . . 6 (x, y, z ((RS) F) ↔ (x dom F ((Fx)Ry (Fx)Sz)))
47 oteltxp 5782 . . . . . 6 (x, y, z ((R F) ⊗ (S F)) ↔ (x, y (R F) x, z (S F)))
4832, 46, 473bitr4i 268 . . . . 5 (x, y, z ((RS) F) ↔ x, y, z ((R F) ⊗ (S F)))
49 opeq2 4579 . . . . . . 7 (t = y, zx, t = x, y, z)
5049eleq1d 2419 . . . . . 6 (t = y, z → (x, t ((RS) F) ↔ x, y, z ((RS) F)))
5149eleq1d 2419 . . . . . 6 (t = y, z → (x, t ((R F) ⊗ (S F)) ↔ x, y, z ((R F) ⊗ (S F))))
5250, 51bibi12d 312 . . . . 5 (t = y, z → ((x, t ((RS) F) ↔ x, t ((R F) ⊗ (S F))) ↔ (x, y, z ((RS) F) ↔ x, y, z ((R F) ⊗ (S F)))))
5348, 52mpbiri 224 . . . 4 (t = y, z → (x, t ((RS) F) ↔ x, t ((R F) ⊗ (S F))))
5453exlimivv 1635 . . 3 (yz t = y, z → (x, t ((RS) F) ↔ x, t ((R F) ⊗ (S F))))
553, 54ax-mp 5 . 2 (x, t ((RS) F) ↔ x, t ((R F) ⊗ (S F)))
5655eqrelriv 4850 1 ((RS) F) = ((R F) ⊗ (S F))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561   class class class wbr 4639   ∘ ccom 4721  dom cdm 4772  Fun wfun 4775   ‘cfv 4781   ⊗ ctxp 5735 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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-2nd 4797  df-txp 5736 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator