Theorem opabiotafun 6421

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 6084	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦})
2		mo2icl 3526	. . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
3		unieq 4596	. . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧})
4		vex 3343	. . . . . . 7 ⊢ 𝑧 ∈ V
5	4	unisn 4603	. . . . . 6 ⊢ ∪ {𝑧} = 𝑧
6	3, 5	syl6req 2811	. . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑})
7	2, 6	mpg 1873	. . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}
8		nfv 1992	. . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦}
9		nfab1 2904	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
10	9	nfeq1 2916	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧}
11		sneq 4331	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
12	11	eqeq2d 2770	. . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
13	8, 10, 12	cbvmo 2644	. . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
14	7, 13	mpbir 221	. . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}
15	1, 14	mpgbir 1875	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
16		opabiota.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
17	16	funeqi 6070	. 2 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
18	15, 17	mpbir 221	1 ⊢ Fun 𝐹

Syntax hints:   → wi 4   = wceq 1632  ∃*wmo 2608  {cab 2746  {csn 4321  ∪ cuni 4588  {copab 4864  Fun wfun 6043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-fun 6051

This theorem is referenced by:  opabiota  6423