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Theorem domfnex 5870
 Description: The domain function is stratified. (Contributed by Scott Fenton, 9-Aug-2019.)
Assertion
Ref Expression
domfnex Dom V

Proof of Theorem domfnex
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-domfn 5770 . . 3 Dom = (x V dom x)
2 elin 3219 . . . . . . . . 9 ({w}, {z}, {y}, x ( Ins4 SI3 Swap Ins2 Ins2 S ) ↔ ({w}, {z}, {y}, x Ins4 SI3 Swap {w}, {z}, {y}, x Ins2 Ins2 S ))
3 vex 2862 . . . . . . . . . . . 12 x V
43oqelins4 5794 . . . . . . . . . . 11 ({w}, {z}, {y}, x Ins4 SI3 Swap {w}, {z}, {y} SI3 Swap )
5 vex 2862 . . . . . . . . . . . . 13 w V
6 vex 2862 . . . . . . . . . . . . 13 z V
7 vex 2862 . . . . . . . . . . . . 13 y V
85, 6, 7otsnelsi3 5805 . . . . . . . . . . . 12 ({w}, {z}, {y} SI3 Swap w, z, y Swap )
9 df-br 4640 . . . . . . . . . . . 12 (w Swap z, yw, z, y Swap )
106, 7brswap2 4860 . . . . . . . . . . . 12 (w Swap z, yw = y, z)
118, 9, 103bitr2i 264 . . . . . . . . . . 11 ({w}, {z}, {y} SI3 Swap w = y, z)
124, 11bitri 240 . . . . . . . . . 10 ({w}, {z}, {y}, x Ins4 SI3 Swap w = y, z)
13 snex 4111 . . . . . . . . . . . 12 {z} V
1413otelins2 5791 . . . . . . . . . . 11 ({w}, {z}, {y}, x Ins2 Ins2 S {w}, {y}, x Ins2 S )
15 snex 4111 . . . . . . . . . . . . 13 {y} V
1615otelins2 5791 . . . . . . . . . . . 12 ({w}, {y}, x Ins2 S {w}, x S )
175, 3opelssetsn 4760 . . . . . . . . . . . 12 ({w}, x S w x)
1816, 17bitri 240 . . . . . . . . . . 11 ({w}, {y}, x Ins2 S w x)
1914, 18bitri 240 . . . . . . . . . 10 ({w}, {z}, {y}, x Ins2 Ins2 S w x)
2012, 19anbi12i 678 . . . . . . . . 9 (({w}, {z}, {y}, x Ins4 SI3 Swap {w}, {z}, {y}, x Ins2 Ins2 S ) ↔ (w = y, z w x))
212, 20bitri 240 . . . . . . . 8 ({w}, {z}, {y}, x ( Ins4 SI3 Swap Ins2 Ins2 S ) ↔ (w = y, z w x))
2221exbii 1582 . . . . . . 7 (w{w}, {z}, {y}, x ( Ins4 SI3 Swap Ins2 Ins2 S ) ↔ w(w = y, z w x))
23 elima1c 4947 . . . . . . 7 ({z}, {y}, x (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ↔ w{w}, {z}, {y}, x ( Ins4 SI3 Swap Ins2 Ins2 S ))
24 df-clel 2349 . . . . . . 7 (y, z xw(w = y, z w x))
2522, 23, 243bitr4i 268 . . . . . 6 ({z}, {y}, x (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ↔ y, z x)
2625exbii 1582 . . . . 5 (z{z}, {y}, x (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ↔ zy, z x)
27 elima1c 4947 . . . . 5 ({y}, x ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) “ 1c) ↔ z{z}, {y}, x (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c))
28 eldm2 4899 . . . . 5 (y dom xzy, z x)
2926, 27, 283bitr4i 268 . . . 4 ({y}, x ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) “ 1c) ↔ y dom x)
3029releqmpt 5808 . . 3 ((V × V) ∩ ∼ (( Ins3 S Ins2 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) “ 1c)) “ 1c)) = (x V dom x)
311, 30eqtr4i 2376 . 2 Dom = ((V × V) ∩ ∼ (( Ins3 S Ins2 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) “ 1c)) “ 1c))
32 vvex 4109 . . 3 V V
33 swapex 4742 . . . . . . . 8 Swap V
3433si3ex 5806 . . . . . . 7 SI3 Swap V
3534ins4ex 5799 . . . . . 6 Ins4 SI3 Swap V
36 ssetex 4744 . . . . . . . 8 S V
3736ins2ex 5797 . . . . . . 7 Ins2 S V
3837ins2ex 5797 . . . . . 6 Ins2 Ins2 S V
3935, 38inex 4105 . . . . 5 ( Ins4 SI3 Swap Ins2 Ins2 S ) V
40 1cex 4142 . . . . 5 1c V
4139, 40imaex 4747 . . . 4 (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) V
4241, 40imaex 4747 . . 3 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) “ 1c) V
4332, 42mptexlem 5810 . 2 ((V × V) ∩ ∼ (( Ins3 S Ins2 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) “ 1c)) “ 1c)) V
4431, 43eqeltri 2423 1 Dom V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208   ⊕ csymdif 3209  {csn 3737  1cc1c 4134  ⟨cop 4561   class class class wbr 4639   Swap cswap 4718   S csset 4719   “ cima 4722   × cxp 4770  ◡ccnv 4771  dom cdm 4772   ↦ cmpt 5651   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757   Dom cdomfn 5769 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-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-2nd 4797  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758  df-domfn 5770 This theorem is referenced by: (None)
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