Theorem dmmptdf2 39439

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
dmmptdf2.x	⊢ Ⅎ𝑥𝜑
dmmptdf2.b	⊢ Ⅎ𝑥𝐵
dmmptdf2.a	⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)
dmmptdf2.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
dmmptdf2	⊢ (𝜑 → dom 𝐴 = 𝐵)

Proof of Theorem dmmptdf2

Step	Hyp	Ref	Expression
1		dmmptdf2.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		dmmptdf2.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
3		elex 3212	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V)
5	4	ex 450	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐶 ∈ V))
6	1, 5	ralrimi 2957	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
7		dmmptdf2.b	. . . 4 ⊢ Ⅎ𝑥𝐵
8	7	rabid2f 3119	. . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
9	6, 8	sylibr 224	. 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V})
10		dmmptdf2.a	. . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)
11	10	dmmpt 5630	. 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}
12	9, 11	syl6reqr 2675	1 ⊢ (𝜑 → dom 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912  {crab 2916  Vcvv 3200   ↦ cmpt 4729  dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)