Theorem opabiotafun 6217

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 5883	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦})
2		mo2icl 3372	. . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
3		unieq 4415	. . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧})
4		vex 3194	. . . . . . 7 ⊢ 𝑧 ∈ V
5	4	unisn 4422	. . . . . 6 ⊢ ∪ {𝑧} = 𝑧
6	3, 5	syl6req 2677	. . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑})
7	2, 6	mpg 1721	. . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}
8		nfv 1845	. . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦}
9		nfab1 2769	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
10	9	nfeq1 2780	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧}
11		sneq 4163	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
12	11	eqeq2d 2636	. . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
13	8, 10, 12	cbvmo 2510	. . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
14	7, 13	mpbir 221	. . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}
15	1, 14	mpgbir 1723	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
16		opabiota.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
17	16	funeqi 5870	. 2 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
18	15, 17	mpbir 221	1 ⊢ Fun 𝐹

Syntax hints:   → wi 4   = wceq 1480  ∃*wmo 2475  {cab 2612  {csn 4153  ∪ cuni 4407  {copab 4677  Fun wfun 5844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-fun 5852

This theorem is referenced by:  opabiota  6219