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Theorem fvfullfunlem1 5861
 Description: Lemma for fvfullfun 5864. Calculate the domain of part one of the full function definition. (Contributed by SF, 9-Mar-2015.)
Assertion
Ref Expression
fvfullfunlem1 dom (( I F) ( ∼ I F)) = {x ∃!y xFy}
Distinct variable group:   x,F,y

Proof of Theorem fvfullfunlem1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eldm 4898 . . 3 (x dom (( I F) ( ∼ I F)) ↔ y x(( I F) ( ∼ I F))y)
2 fnfullfunlem1 5856 . . . 4 (x(( I F) ( ∼ I F))y ↔ (xFy z(xFzz = y)))
32exbii 1582 . . 3 (y x(( I F) ( ∼ I F))yy(xFy z(xFzz = y)))
4 nfv 1619 . . . . 5 z xFy
54eu1 2225 . . . 4 (∃!y xFyy(xFy z([z / y]xFyy = z)))
6 nfv 1619 . . . . . . . . 9 y xFz
7 breq2 4643 . . . . . . . . 9 (y = z → (xFyxFz))
86, 7sbie 2038 . . . . . . . 8 ([z / y]xFyxFz)
9 equcom 1680 . . . . . . . 8 (y = zz = y)
108, 9imbi12i 316 . . . . . . 7 (([z / y]xFyy = z) ↔ (xFzz = y))
1110albii 1566 . . . . . 6 (z([z / y]xFyy = z) ↔ z(xFzz = y))
1211anbi2i 675 . . . . 5 ((xFy z([z / y]xFyy = z)) ↔ (xFy z(xFzz = y)))
1312exbii 1582 . . . 4 (y(xFy z([z / y]xFyy = z)) ↔ y(xFy z(xFzz = y)))
145, 13bitr2i 241 . . 3 (y(xFy z(xFzz = y)) ↔ ∃!y xFy)
151, 3, 143bitri 262 . 2 (x dom (( I F) ( ∼ I F)) ↔ ∃!y xFy)
1615abbi2i 2464 1 dom (( I F) ( ∼ I F)) = {x ∃!y xFy}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  {cab 2339   ∼ ccompl 3205   ∖ cdif 3206   class class class wbr 4639   ∘ ccom 4721   I cid 4763  dom cdm 4772 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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by:  fvfullfunlem3  5863  fvfullfun  5864
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