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 ∈ A ↦ B) = A)

Distinct variable group:   x,A

Allowed substitution hints:   B(x)   𝑉(x)

Proof of Theorem dmmptg

Step	Hyp	Ref	Expression
1		elex 2560	. . . 4 ⊢ (B ∈ 𝑉 → B ∈ V)
2	1	ralimi 2378	. . 3 ⊢ (∀x ∈ A B ∈ 𝑉 → ∀x ∈ A B ∈ V)
3		rabid2 2480	. . 3 ⊢ (A = {x ∈ A ∣ B ∈ V} ↔ ∀x ∈ A B ∈ V)
4	2, 3	sylibr 137	. 2 ⊢ (∀x ∈ A B ∈ 𝑉 → A = {x ∈ A ∣ B ∈ V})
5		eqid 2037	. . 3 ⊢ (x ∈ A ↦ B) = (x ∈ A ↦ B)
6	5	dmmpt 4759	. 2 ⊢ dom (x ∈ A ↦ B) = {x ∈ A ∣ B ∈ V}
7	4, 6	syl6reqr 2088	1 ⊢ (∀x ∈ A B ∈ 𝑉 → dom (x ∈ A ↦ B) = A)

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