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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)
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