Theorem mptrcl 6777

Description: Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.)

Hypothesis

Ref	Expression
mptrcl.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
mptrcl	⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝑋(𝑥)

Proof of Theorem mptrcl

Step	Hyp	Ref	Expression
1		n0i 4299	. 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → ¬ (𝐹‘𝑋) = ∅)
2		mptrcl.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	dmmptss 6095	. . . 4 ⊢ dom 𝐹 ⊆ 𝐴
4	3	sseli 3963	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝐴)
5		ndmfv 6700	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
6	4, 5	nsyl4 161	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ 𝐴)
7	1, 6	syl 17	1 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2114  ∅c0 4291   ↦ cmpt 5146  dom cdm 5555  ‘cfv 6355

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-xp 5561  df-rel 5562  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fv 6363

This theorem is referenced by:  bitsval  15773  subcrcl  17086  initorcl  17254  termorcl  17255  zeroorcl  17256  submrcl  17967  issubg  18279  isnsg  18307  issubrg  19535  issdrg  19574  abvrcl  19592  isobs  20864  islocfin  22125  kgeni  22145  elmptrab  22435  isphtpc  23598  cfili  23871  cfilfcls  23877  plybss  24784  eleenn  26682