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Theorem dmmptg 4761
Description: The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
Assertion
Ref Expression
dmmptg (x A B 𝑉 → dom (x AB) = A)
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   𝑉(x)

Proof of Theorem dmmptg
StepHypRef Expression
1 elex 2560 . . . 4 (B 𝑉B V)
21ralimi 2378 . . 3 (x A B 𝑉x A B V)
3 rabid2 2480 . . 3 (A = {x AB V} ↔ x A B V)
42, 3sylibr 137 . 2 (x A B 𝑉A = {x AB V})
5 eqid 2037 . . 3 (x AB) = (x AB)
65dmmpt 4759 . 2 dom (x AB) = {x AB V}
74, 6syl6reqr 2088 1 (x A B 𝑉 → dom (x AB) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wral 2300  {crab 2304  Vcvv 2551  cmpt 3809  dom cdm 4288
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-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
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-rab 2309  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-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  resfunexg  5325  rdgtfr  5901  rdgruledefgg  5902
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