Theorem fnoprabg 7275

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 2663	. . . . . 6 ⊢ (∃!𝑧𝜓 → ∃*𝑧𝜓)
2	1	imim2i 16	. . . . 5 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃*𝑧𝜓))
3		moanimv 2704	. . . . 5 ⊢ (∃*𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑧𝜓))
4	2, 3	sylibr 236	. . . 4 ⊢ ((𝜑 → ∃!𝑧𝜓) → ∃*𝑧(𝜑 ∧ 𝜓))
5	4	2alimi 1813	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → ∀𝑥∀𝑦∃*𝑧(𝜑 ∧ 𝜓))
6		funoprabg 7273	. . 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 7255	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑 ∧ 𝜓)}
9		nfa1 2155	. . . 4 ⊢ Ⅎ𝑥∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
10		nfa2 2176	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
11		simpl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
12	11	exlimiv 1931	. . . . . . 7 ⊢ (∃𝑧(𝜑 ∧ 𝜓) → 𝜑)
13		euex 2662	. . . . . . . . . 10 ⊢ (∃!𝑧𝜓 → ∃𝑧𝜓)
14	13	imim2i 16	. . . . . . . . 9 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧𝜓))
15	14	ancld 553	. . . . . . . 8 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → (𝜑 ∧ ∃𝑧𝜓)))
16		19.42v 1954	. . . . . . . 8 ⊢ (∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑧𝜓))
17	15, 16	syl6ibr 254	. . . . . . 7 ⊢ ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧(𝜑 ∧ 𝜓)))
18	12, 17	impbid2 228	. . . . . 6 ⊢ ((𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
19	18	sps 2184	. . . . 5 ⊢ (∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
20	19	sps 2184	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑 ∧ 𝜓) ↔ 𝜑))
21	9, 10, 20	opabbid 5131	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
22	8, 21	syl5eq 2868	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
23		df-fn 6358	. 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 585	1 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780  ∃*wmo 2620  ∃!weu 2653  {copab 5128  dom cdm 5555  Fun wfun 6349   Fn wfn 6350  {coprab 7157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-fun 6357  df-fn 6358  df-oprab 7160

This theorem is referenced by:  fnoprab  7277  ovg  7313