Theorem opabiotafun 6749

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 6374	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦})
2		mo2icl 3613	. . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
3		unieq 4807	. . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧})
4		vex 3402	. . . . . . 7 ⊢ 𝑧 ∈ V
5	4	unisn 4818	. . . . . 6 ⊢ ∪ {𝑧} = 𝑧
6	3, 5	eqtr2di 2790	. . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑})
7	2, 6	mpg 1804	. . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}
8		nfv 1921	. . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦}
9		nfab1 2901	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
10	9	nfeq1 2914	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧}
11		sneq 4526	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
12	11	eqeq2d 2749	. . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
13	8, 10, 12	cbvmow 2604	. . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
14	7, 13	mpbir 234	. . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}
15	1, 14	mpgbir 1806	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
16		opabiota.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
17	16	funeqi 6360	. 2 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
18	15, 17	mpbir 234	1 ⊢ Fun 𝐹

Syntax hints:   → wi 4   = wceq 1542  ∃*wmo 2538  {cab 2716  {csn 4516  ∪ cuni 4796  {copab 5092  Fun wfun 6333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-fun 6341

This theorem is referenced by:  opabiota  6751