Theorem fmptd2f 45593

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 2737	. 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 1785   ∈ wcel 2114   ↦ cmpt 5181  ⟶wf 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by:  climinf2mpt  46072  climinfmpt  46073  limsupvaluzmpt  46075  limsupre2mpt  46088  limsupre3mpt  46092  limsupreuzmpt  46097  supcnvlimsupmpt  46099  liminfvalxrmpt  46144  liminflbuz2  46173  dvnprodlem1  46304  sge0z  46733  sge0f1o  46740  smfsupmpt  47173  smfinfmpt  47177  smflimsupmpt  47187  smfliminfmpt  47190  smfsupdmmbllem  47202  smfinfdmmbllem  47206