Theorem fnopabg 5206

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Hypothesis

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

Assertion

Ref	Expression
fnopabg	⊢ (∀x ∈ A ∃!yφ ↔ F Fn A)

Distinct variable group:   x,y,A

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

Proof of Theorem fnopabg

Step	Hyp	Ref	Expression
1		moanimv 2262	. . . . . 6 ⊢ (∃*y(x ∈ A ∧ φ) ↔ (x ∈ A → ∃*yφ))
2	1	albii 1566	. . . . 5 ⊢ (∀x∃*y(x ∈ A ∧ φ) ↔ ∀x(x ∈ A → ∃*yφ))
3		funopab 5140	. . . . 5 ⊢ (Fun {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} ↔ ∀x∃*y(x ∈ A ∧ φ))
4		df-ral 2620	. . . . 5 ⊢ (∀x ∈ A ∃*yφ ↔ ∀x(x ∈ A → ∃*yφ))
5	2, 3, 4	3bitr4ri 269	. . . 4 ⊢ (∀x ∈ A ∃*yφ ↔ Fun {⟨x, y⟩ ∣ (x ∈ A ∧ φ)})
6		dmopab3 4918	. . . 4 ⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} = A)
7	5, 6	anbi12i 678	. . 3 ⊢ ((∀x ∈ A ∃*yφ ∧ ∀x ∈ A ∃yφ) ↔ (Fun {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} ∧ dom {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} = A))
8		r19.26 2747	. . 3 ⊢ (∀x ∈ A (∃*yφ ∧ ∃yφ) ↔ (∀x ∈ A ∃*yφ ∧ ∀x ∈ A ∃yφ))
9		df-fn 4791	. . 3 ⊢ ({⟨x, y⟩ ∣ (x ∈ A ∧ φ)} Fn A ↔ (Fun {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} ∧ dom {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} = A))
10	7, 8, 9	3bitr4i 268	. 2 ⊢ (∀x ∈ A (∃*yφ ∧ ∃yφ) ↔ {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} Fn A)
11		eu5 2242	. . . 4 ⊢ (∃!yφ ↔ (∃yφ ∧ ∃*yφ))
12		ancom 437	. . . 4 ⊢ ((∃yφ ∧ ∃*yφ) ↔ (∃*yφ ∧ ∃yφ))
13	11, 12	bitri 240	. . 3 ⊢ (∃!yφ ↔ (∃*yφ ∧ ∃yφ))
14	13	ralbii 2639	. 2 ⊢ (∀x ∈ A ∃!yφ ↔ ∀x ∈ A (∃*yφ ∧ ∃yφ))
15		fnopabg.1	. . 3 ⊢ F = {⟨x, y⟩ ∣ (x ∈ A ∧ φ)}
16	15	fneq1i 5179	. 2 ⊢ (F Fn A ↔ {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} Fn A)
17	10, 14, 16	3bitr4i 268	1 ⊢ (∀x ∈ A ∃!yφ ↔ F Fn A)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2615  {copab 4623  dom cdm 4773  Fun wfun 4776   Fn wfn 4777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791

This theorem is referenced by:  fnopab2g  5207  fnopab  5208