Theorem fnoprabg 7539

Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)

Assertion

Ref	Expression
fnoprabg	⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦, 𝑧)

Proof of Theorem fnoprabg

Step	Hyp	Ref	Expression
1		eumo 2608	. . . . . 6 ⊢ (∃!𝑧𝜓 → ∃*𝑧𝜓)
2	1	imim2i 17	. . . . 5 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃*𝑧𝜓))
3		moanimv 2649	. . . . 5 ⊢ (∃*𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑧𝜓))
4	2, 3	sylibr 237	. . . 4 ⊢ ((𝜑 → ∃!𝑧𝜓) → ∃*𝑧(𝜑 ∧ 𝜓))
5	4	2alimi 1845	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → ∀𝑥∀𝑦∃*𝑧(𝜑 ∧ 𝜓))
6		funoprabg 7537	. . 3 ⊢ (∀𝑥∀𝑦∃*𝑧(𝜑 ∧ 𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)})
7	5, 6	syl 18	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)})
8		dmoprab 7519	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑 ∧ 𝜓)}
9		nfa1 2189	. . . 4 ⊢ Ⅎ𝑥∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
10		nfa2 2213	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
11		simpl 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
12	11	exlimiv 1963	. . . . . . 7 ⊢ (∃𝑧(𝜑 ∧ 𝜓) → 𝜑)
13		euex 2607	. . . . . . . . . 10 ⊢ (∃!𝑧𝜓 → ∃𝑧𝜓)
14	13	imim2i 17	. . . . . . . . 9 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧𝜓))
15	14	ancld 560	. . . . . . . 8 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → (𝜑 ∧ ∃𝑧𝜓)))
16		19.42v 1986	. . . . . . . 8 ⊢ (∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑧𝜓))
17	15, 16	imbitrrdi 255	. . . . . . 7 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧(𝜑 ∧ 𝜓)))
18	12, 17	impbid2 229	. . . . . 6 ⊢ ((𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
19	18	sps 2224	. . . . 5 ⊢ (∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
20	19	sps 2224	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
21	9, 10, 20	opabbid 5178	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
22	8, 21	eqtrid 2812	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
23		df-fn 6543	. 2 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} ∧ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
24	7, 22, 23	sylanbrc 595	1 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812  ∃*wmo 2567  ∃!weu 2598  {copab 5175  dom cdm 5663  Fun wfun 6534   Fn wfn 6535  {coprab 7417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-fun 6542  df-fn 6543  df-oprab 7420

This theorem is used by:  fnoprab  7541  ovg  7581