Theorem mptex2 6384

Description: If a class given as a map-to notation is a set, it's image values are set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
mptex2.1	⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)

Assertion

Ref	Expression
mptex2	⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Distinct variable groups:   𝑡,𝐴   𝑡,𝐶

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

Proof of Theorem mptex2

Step	Hyp	Ref	Expression
1		mptex2.1	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
2		eqid 2622	. . . 4 ⊢ (𝑡 ∈ 𝐴 ↦ 𝐵) = (𝑡 ∈ 𝐴 ↦ 𝐵)
3	2	fmpt 6381	. . 3 ⊢ (∀𝑡 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ (𝑡 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
4	1, 3	sylibr 224	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐴 𝐵 ∈ 𝐶)
5	4	r19.21bi 2932	1 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ∀wral 2912   ↦ cmpt 4729  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  divcncf  23216  cncfcompt  40096  cncficcgt0  40101  cncfcompt2  40112  itgsubsticclem  40191  sge0iunmptlemre  40632  hoicvrrex  40770  smfadd  40973