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Theorem brdomg 6165
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.)
Assertion
Ref Expression
brdomg (B 𝐶 → (ABf f:A1-1B))
Distinct variable groups:   A,f   B,f
Allowed substitution hint:   𝐶(f)

Proof of Theorem brdomg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 6162 . . . 4 Rel ≼
21brrelexi 4327 . . 3 (ABA V)
32a1i 9 . 2 (B 𝐶 → (ABA V))
4 f1f 5035 . . . . 5 (f:A1-1Bf:AB)
5 fdm 4993 . . . . . 6 (f:AB → dom f = A)
6 vex 2554 . . . . . . 7 f V
76dmex 4541 . . . . . 6 dom f V
85, 7syl6eqelr 2126 . . . . 5 (f:ABA V)
94, 8syl 14 . . . 4 (f:A1-1BA V)
109exlimiv 1486 . . 3 (f f:A1-1BA V)
1110a1i 9 . 2 (B 𝐶 → (f f:A1-1BA V))
12 f1eq2 5031 . . . . 5 (x = A → (f:x1-1yf:A1-1y))
1312exbidv 1703 . . . 4 (x = A → (f f:x1-1yf f:A1-1y))
14 f1eq3 5032 . . . . 5 (y = B → (f:A1-1yf:A1-1B))
1514exbidv 1703 . . . 4 (y = B → (f f:A1-1yf f:A1-1B))
16 df-dom 6159 . . . 4 ≼ = {⟨x, y⟩ ∣ f f:x1-1y}
1713, 15, 16brabg 3997 . . 3 ((A V B 𝐶) → (ABf f:A1-1B))
1817expcom 109 . 2 (B 𝐶 → (A V → (ABf f:A1-1B)))
193, 11, 18pm5.21ndd 620 1 (B 𝐶 → (ABf f:A1-1B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551   class class class wbr 3755  dom cdm 4288  wf 4841  1-1wf1 4842  cdom 6156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-fn 4848  df-f 4849  df-f1 4850  df-dom 6159
This theorem is referenced by:  brdomi  6166  brdom  6167  f1dom2g  6172  f1domg  6174  dom3d  6190
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