Theorem dmmptdf 45225

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	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcv 2892	. 2 ⊢ Ⅎ𝑥𝐵
3		dmmptdf.a	. 2 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)
4		dmmptdf.c	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
5	1, 2, 3, 4	dmmptdff 45224	1 ⊢ (𝜑 → dom 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109   ↦ cmpt 5191  dom cdm 5641

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-mpt 5192  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654

This theorem is referenced by:  smfpimltmpt  46751  smfadd  46770  smfpimgtmpt  46786  smfpimioompt  46791  smfrec  46794  smfmul  46800  smfmulc1  46801  smfsupmpt  46820  smfinfmpt  46824  smflimsupmpt  46834  smfliminfmpt  46837