Theorem dmmptdf 38412

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
dmmptdf.x	⊢ Ⅎ𝑥𝜑
dmmptdf.a	⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)
dmmptdf.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
dmmptdf	⊢ (𝜑 → dom 𝐴 = 𝐵)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem dmmptdf

Step	Hyp	Ref	Expression
1		dmmptdf.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		dmmptdf.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
3		elex 3185	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V)
5	4	ex 449	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐶 ∈ V))
6	1, 5	ralrimi 2940	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
7		rabid2 3096	. . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
8	6, 7	sylibr 223	. 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V})
9		dmmptdf.a	. . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)
10	9	dmmpt 5547	. 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}
11	8, 10	syl6reqr 2663	1 ⊢ (𝜑 → dom 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ↦ cmpt 4643  dom cdm 5038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by:  smfpimltmpt  39633  smfpimltxrmpt  39645  smfadd  39651  smfpimgtmpt  39667  smfpimgtxrmpt  39670  smfpimioompt  39671  smfrec  39674  smfmul  39680  smfmulc1  39681