Theorem dmmpt2g 7506

Description: Domain of an operation given by the maps-to notation, closed form of dmmpt2 7503. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)

Hypothesis

Ref	Expression
dmmpt2g.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
dmmpt2g	⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))

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

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem dmmpt2g

Step	Hyp	Ref	Expression
1		simpl 476	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
2	1	ralrimivva 3180	. 2 ⊢ (𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉)
3		dmmpt2g.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	dmmpt2ga 7505	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
5	2, 4	syl 17	1 ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117   × cxp 5340  dom cdm 5342   ↦ 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:  aovmpt4g  42103