Theorem opabiotafun 6961

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 6571	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦})
2		mo2icl 3677	. . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
3		unieq 4883	. . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧})
4		unisnv 4892	. . . . . 6 ⊢ ∪ {𝑧} = 𝑧
5	3, 4	eqtr2di 2815	. . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑})
6	2, 5	mpg 1827	. . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}
7		nfv 1944	. . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦}
8		nfab1 2927	. . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
9	8	nfeq1 2940	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧}
10		sneq 4599	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
11	10	eqeq2d 2774	. . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
12	7, 9, 11	cbvmow 2631	. . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧})
13	6, 12	mpbir 234	. . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}
14	1, 13	mpgbir 1829	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
15		opabiota.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
16	15	funeqi 6557	. 2 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
17	14, 16	mpbir 234	1 ⊢ Fun 𝐹

Syntax hints:   → wi 4   = wceq 1570  ∃*wmo 2565  {cab 2741  {csn 4589  ∪ cuni 4872  {copab 5173  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-fun 6538

This theorem is referenced by:  opabiota  6963