Theorem fnopabg 6637

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 2612	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
2	1	albii 1819	. . . . 5 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
3		funopab 6535	. . . . 5 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-ral 3045	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
5	2, 3, 4	3bitr4ri 304	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6		dmopab3 5873	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
7	5, 6	anbi12i 628	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴))
8		r19.26 3091	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑))
9		df-fn 6502	. . 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 303	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
11		df-eu 2562	. . . 4 ⊢ (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
12	11	biancomi 462	. . 3 ⊢ (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
13	12	ralbii 3075	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
14		fnopabg.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
15	14	fneq1i 6597	. 2 ⊢ (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
16	10, 13, 15	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∃!weu 2561  ∀wral 3044  {copab 5164  dom cdm 5631  Fun wfun 6493   Fn wfn 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6501  df-fn 6502

This theorem is referenced by:  fnopab  6638  mptfng  6639  axcontlem2  28945  tfsconcatfn  43320  tfsconcatfv1  43321  tfsconcatfv2  43322