Theorem dmmpt2 7503

Description: Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)

Hypotheses

Ref	Expression
fmpt2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
fnmpt2i.2	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
dmmpt2	⊢ dom 𝐹 = (𝐴 × 𝐵)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpt2

Step	Hyp	Ref	Expression
1		fmpt2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		fnmpt2i.2	. . 3 ⊢ 𝐶 ∈ V
3	1, 2	fnmpt2i 7502	. 2 ⊢ 𝐹 Fn (𝐴 × 𝐵)
4		fndm 6223	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
5	3, 4	ax-mp 5	1 ⊢ dom 𝐹 = (𝐴 × 𝐵)

Syntax hints:   = wceq 1658   ∈ wcel 2166  Vcvv 3414   × cxp 5340  dom cdm 5342   Fn wfn 6118   ↦ cmpt2 6907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429

This theorem is referenced by:  1div0  11011  swrd00  13704  swrd0  13723  pfx00  13753  pfx0  13754  repsundef  13887  cshnz  13911  cshnzOLD  13912  imasvscafn  16550  imasvscaval  16551  iscnp2  21414  xkococnlem  21833  ucnima  22455  ucnprima  22456  tngtopn  22824  1div0apr  27882  smatlem  30408  elunirnmbfm  30860  rrxsphere  43300