Theorem mptrcl 7001

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 4294	. 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → ¬ (𝐹‘𝑋) = ∅)
2		mptrcl.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	dmmptss 6244	. . . 4 ⊢ dom 𝐹 ⊆ 𝐴
4	3	sseli 3934	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝐴)
5		ndmfv 6915	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
6	4, 5	nsyl4 159	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ 𝐴)
7	1, 6	syl 18	1 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  ∅c0 4287   ↦ cmpt 5193  dom cdm 5663  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fv 6546

This theorem is referenced by:  bitsval  16483  subcrcl  17874  initorcl  18048  termorcl  18049  zeroorcl  18050  submrcl  18861  issubg  19193  isnsg  19222  issubrng  20633  issubrg  20657  issdrg  20872  abvrcl  20897  isobs  21851  mhprcl  22287  islocfin  23655  kgeni  23675  elmptrab  23965  isphtpc  25134  cfili  25408  cfilfcls  25414  plybss  26332  eleenn  29224  neircl  49660  sectrcl  49777  invrcl  49779  isorcl  49788  sectpropdlem  49791  invpropdlem  49793  isopropdlem  49795  lmdrcl  50406  cmdrcl  50407  lmdfval2  50410  cmdfval2  50411