Theorem fnoprabg 7491

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 2579	. . . . . 6 ⊢ (∃!𝑧𝜓 → ∃*𝑧𝜓)
2	1	imim2i 16	. . . . 5 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃*𝑧𝜓))
3		moanimv 2620	. . . . 5 ⊢ (∃*𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑧𝜓))
4	2, 3	sylibr 234	. . . 4 ⊢ ((𝜑 → ∃!𝑧𝜓) → ∃*𝑧(𝜑 ∧ 𝜓))
5	4	2alimi 1814	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → ∀𝑥∀𝑦∃*𝑧(𝜑 ∧ 𝜓))
6		funoprabg 7489	. . 3 ⊢ (∀𝑥∀𝑦∃*𝑧(𝜑 ∧ 𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)})
7	5, 6	syl 17	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)})
8		dmoprab 7471	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑 ∧ 𝜓)}
9		nfa1 2157	. . . 4 ⊢ Ⅎ𝑥∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
10		nfa2 2182	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
11		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
12	11	exlimiv 1932	. . . . . . 7 ⊢ (∃𝑧(𝜑 ∧ 𝜓) → 𝜑)
13		euex 2578	. . . . . . . . . 10 ⊢ (∃!𝑧𝜓 → ∃𝑧𝜓)
14	13	imim2i 16	. . . . . . . . 9 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧𝜓))
15	14	ancld 550	. . . . . . . 8 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → (𝜑 ∧ ∃𝑧𝜓)))
16		19.42v 1955	. . . . . . . 8 ⊢ (∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑧𝜓))
17	15, 16	imbitrrdi 252	. . . . . . 7 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧(𝜑 ∧ 𝜓)))
18	12, 17	impbid2 226	. . . . . 6 ⊢ ((𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
19	18	sps 2193	. . . . 5 ⊢ (∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
20	19	sps 2193	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
21	9, 10, 20	opabbid 5165	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
22	8, 21	eqtrid 2784	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
23		df-fn 6503	. 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 584	1 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781  ∃*wmo 2538  ∃!weu 2569  {copab 5162  dom cdm 5632  Fun wfun 6494   Fn wfn 6495  {coprab 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-fun 6502  df-fn 6503  df-oprab 7372

This theorem is referenced by:  fnoprab  7493  ovg  7533