Theorem mptrcl 7025

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 4340	. 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → ¬ (𝐹‘𝑋) = ∅)
2		mptrcl.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	dmmptss 6261	. . . 4 ⊢ dom 𝐹 ⊆ 𝐴
4	3	sseli 3979	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝐴)
5		ndmfv 6941	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
6	4, 5	nsyl4 158	. 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ 𝐴)
7	1, 6	syl 17	1 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  ∅c0 4333   ↦ cmpt 5225  dom cdm 5685  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569

This theorem is referenced by:  bitsval  16461  subcrcl  17860  initorcl  18035  termorcl  18036  zeroorcl  18037  submrcl  18815  issubg  19144  isnsg  19173  issubrng  20547  issubrg  20571  issdrg  20789  abvrcl  20814  isobs  21740  mhprcl  22147  islocfin  23525  kgeni  23545  elmptrab  23835  isphtpc  25026  cfili  25302  cfilfcls  25308  plybss  26233  eleenn  28911  neircl  48802