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Theorem dmfco 5381
 Description: Domains of a function composition. (Contributed by set.mm contributors, 27-Jan-1997.)
Assertion
Ref Expression
dmfco ((Fun G A dom G) → (A dom (F G) ↔ (GA) dom F))

Proof of Theorem dmfco
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5339 . . . . 5 (GA) V
2 breq1 4642 . . . . 5 (y = (GA) → (yFz ↔ (GA)Fz))
31, 2ceqsexv 2894 . . . 4 (y(y = (GA) yFz) ↔ (GA)Fz)
4 eqcom 2355 . . . . . . 7 (y = (GA) ↔ (GA) = y)
5 funbrfvb 5360 . . . . . . 7 ((Fun G A dom G) → ((GA) = yAGy))
64, 5syl5bb 248 . . . . . 6 ((Fun G A dom G) → (y = (GA) ↔ AGy))
76anbi1d 685 . . . . 5 ((Fun G A dom G) → ((y = (GA) yFz) ↔ (AGy yFz)))
87exbidv 1626 . . . 4 ((Fun G A dom G) → (y(y = (GA) yFz) ↔ y(AGy yFz)))
93, 8syl5rbbr 251 . . 3 ((Fun G A dom G) → (y(AGy yFz) ↔ (GA)Fz))
109exbidv 1626 . 2 ((Fun G A dom G) → (zy(AGy yFz) ↔ z(GA)Fz))
11 eldm 4898 . . 3 (A dom (F G) ↔ z A(F G)z)
12 brco 4883 . . . 4 (A(F G)zy(AGy yFz))
1312exbii 1582 . . 3 (z A(F G)zzy(AGy yFz))
1411, 13bitri 240 . 2 (A dom (F G) ↔ zy(AGy yFz))
15 eldm 4898 . 2 ((GA) dom Fz(GA)Fz)
1610, 14, 153bitr4g 279 1 ((Fun G A dom G) → (A dom (F G) ↔ (GA) dom F))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   class class class wbr 4639   ∘ ccom 4721  dom cdm 4772  Fun wfun 4775   ‘cfv 4781 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-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 This theorem is referenced by: (None)
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