Theorem dmmptdf 45800

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 2924	. 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 45799	1 ⊢ (𝜑 → dom 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  Ⅎwnf 1803   ∈ wcel 2142   ↦ cmpt 5181  dom cdm 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  smfpimltmpt  47320  smfadd  47339  smfpimgtmpt  47355  smfpimioompt  47360  smfrec  47363  smfmul  47369  smfmulc1  47370  smfsupmpt  47389  smfinfmpt  47393  smflimsupmpt  47403  smfliminfmpt  47406