Theorem dmmptdf 45191

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 2891	. 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 45190	1 ⊢ (𝜑 → dom 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109   ↦ cmpt 5183  dom cdm 5631

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  smfpimltmpt  46717  smfadd  46736  smfpimgtmpt  46752  smfpimioompt  46757  smfrec  46760  smfmul  46766  smfmulc1  46767  smfsupmpt  46786  smfinfmpt  46790  smflimsupmpt  46800  smfliminfmpt  46803