Theorem fvopab4ndm 5803

Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Hypothesis

Ref	Expression
fvopab4ndm.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Assertion

Ref	Expression
fvopab4ndm	⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐵(𝑥, 𝑦)   𝐹(𝑥, 𝑦)

Proof of Theorem fvopab4ndm

Step	Hyp	Ref	Expression
1		fvopab4ndm.1	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2	1	dmeqi 4982	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		dmopabss 4993	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
4	2, 3	eqsstri 3280	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
5	4	sseli 3244	. 2 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ 𝐴)
6	1	relopabiv 4903	. . 3 ⊢ Rel 𝐹
7		relndmfv 5728	. . 3 ⊢ ((Rel 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹‘𝐵) = ∅)
8	6, 7	mpan 428	. 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
9	5, 8	nsyl5 659	1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∅c0 3520  {copab 4191  dom cdm 4774  Rel wrel 4779  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-dm 4784  df-iota 5337  df-fv 5385

This theorem is used by:  fvmptndm  5804