Theorem fmptd2f 45202

Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
fmptd2f.1	⊢ Ⅎ𝑥𝜑
fmptd2f.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
fmptd2f	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fmptd2f

Step	Hyp	Ref	Expression
1		fmptd2f.1	. 2 ⊢ Ⅎ𝑥𝜑
2		fmptd2f.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
3		eqid 2729	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	1, 2, 3	fmptdf 7071	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1783   ∈ wcel 2109   ↦ cmpt 5183  ⟶wf 6495

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-rex 3054  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-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6501  df-fn 6502  df-f 6503

This theorem is referenced by:  climinf2mpt  45685  climinfmpt  45686  limsupvaluzmpt  45688  limsupre2mpt  45701  limsupre3mpt  45705  limsupreuzmpt  45710  supcnvlimsupmpt  45712  liminfvalxrmpt  45757  liminflbuz2  45786  dvnprodlem1  45917  sge0z  46346  sge0f1o  46353  smfsupmpt  46786  smfinfmpt  46790  smflimsupmpt  46800  smfliminfmpt  46803  smfsupdmmbllem  46815  smfinfdmmbllem  46819