Theorem dmmptdf2 41509

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	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
3	2	elexd 3517	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V)
4	1, 3	ralrimia 41404	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
5		dmmptdf2.b	. . . 4 ⊢ Ⅎ𝑥𝐵
6	5	rabid2f 3385	. . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
7	4, 6	sylibr 236	. 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V})
8		dmmptdf2.a	. . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)
9	8	dmmpt 6097	. 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}
10	7, 9	syl6reqr 2878	1 ⊢ (𝜑 → dom 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536  Ⅎwnf 1783   ∈ wcel 2113  Ⅎwnfc 2964  ∀wral 3141  {crab 3145  Vcvv 3497   ↦ cmpt 5149  dom cdm 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-mpt 5150  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571

This theorem is referenced by: (None)