Theorem opabiotafun 6988

Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 19-May-2015.)

Hypothesis

Ref	Expression
opabiota.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}

Assertion

Ref	Expression
opabiotafun	⊢ Fun 𝐹

Distinct variable group:   𝑥,𝑦,𝐹

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabiotafun

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopab 6600	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦})
2		mo2icl 3719	. . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
3		unieq 4917	. . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧})
4		unisnv 4926	. . . . . 6 ⊢ ∪ {𝑧} = 𝑧
5	3, 4	eqtr2di 2793	. . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑})
6	2, 5	mpg 1796	. . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}
7		nfv 1913	. . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦}
8		nfab1 2906	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
9	8	nfeq1 2920	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧}
10		sneq 4635	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
11	10	eqeq2d 2747	. . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
12	7, 9, 11	cbvmow 2602	. . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
13	6, 12	mpbir 231	. . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}
14	1, 13	mpgbir 1798	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
15		opabiota.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
16	15	funeqi 6586	. 2 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
17	14, 16	mpbir 231	1 ⊢ Fun 𝐹

Syntax hints:   → wi 4   = wceq 1539  ∃*wmo 2537  {cab 2713  {csn 4625  ∪ cuni 4906  {copab 5204  Fun wfun 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-fun 6562

This theorem is referenced by:  opabiota  6990