Theorem fmptd2f 45479

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 2736	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	1, 2, 3	fmptdf 7062	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2113   ↦ cmpt 5179  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  climinf2mpt  45958  climinfmpt  45959  limsupvaluzmpt  45961  limsupre2mpt  45974  limsupre3mpt  45978  limsupreuzmpt  45983  supcnvlimsupmpt  45985  liminfvalxrmpt  46030  liminflbuz2  46059  dvnprodlem1  46190  sge0z  46619  sge0f1o  46626  smfsupmpt  47059  smfinfmpt  47063  smflimsupmpt  47073  smfliminfmpt  47076  smfsupdmmbllem  47088  smfinfdmmbllem  47092