Theorem dmmptdf 40158

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 3399	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V)
5	4	ex 402	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐶 ∈ V))
6	1, 5	ralrimi 3137	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
7		rabid2 3299	. . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V)
8	6, 7	sylibr 226	. 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V})
9		dmmptdf.a	. . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)
10	9	dmmpt 5848	. 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}
11	8, 10	syl6reqr 2851	1 ⊢ (𝜑 → dom 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653  Ⅎwnf 1879   ∈ wcel 2157  ∀wral 3088  {crab 3092  Vcvv 3384   ↦ cmpt 4921  dom cdm 5311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rab 3097  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4843  df-opab 4905  df-mpt 4922  df-xp 5317  df-rel 5318  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324

This theorem is referenced by:  smfpimltmpt  41690  smfpimltxrmpt  41702  smfadd  41708  smfpimgtmpt  41724  smfpimgtxrmpt  41727  smfpimioompt  41728  smfrec  41731  smfmul  41737  smfmulc1  41738  smffmpt  41746  smfsupmpt  41756  smfinfmpt  41760  smflimsupmpt  41770  smfliminfmpt  41773