Theorem riotaprop 5415

Description: Properties of a restricted definite description operator. Todo (df-riota 5393 update): can some uses of riota2f 5413 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Hypotheses

Ref	Expression
riotaprop.0	⊢ Ⅎxψ
riotaprop.1	⊢ B = (℩x ∈ A φ)
riotaprop.2	⊢ (x = B → (φ ↔ ψ))

Assertion

Ref	Expression
riotaprop	⊢ (∃!x ∈ A φ → (B ∈ A ∧ ψ))

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   ψ(x)   B(x)

Proof of Theorem riotaprop

Step	Hyp	Ref	Expression
1		riotaprop.1	. . 3 ⊢ B = (℩x ∈ A φ)
2		riotacl 5406	. . 3 ⊢ (∃!x ∈ A φ → (℩x ∈ A φ) ∈ A)
3	1, 2	syl5eqel 2106	. 2 ⊢ (∃!x ∈ A φ → B ∈ A)
4	1	eqcomi 2026	. . . 4 ⊢ (℩x ∈ A φ) = B
5		nfriota1 5399	. . . . . 6 ⊢ Ⅎx(℩x ∈ A φ)
6	1, 5	nfcxfr 2157	. . . . 5 ⊢ ℲxB
7		riotaprop.0	. . . . 5 ⊢ Ⅎxψ
8		riotaprop.2	. . . . 5 ⊢ (x = B → (φ ↔ ψ))
9	6, 7, 8	riota2f 5413	. . . 4 ⊢ ((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ (℩x ∈ A φ) = B))
10	4, 9	mpbiri 157	. . 3 ⊢ ((B ∈ A ∧ ∃!x ∈ A φ) → ψ)
11	3, 10	mpancom 401	. 2 ⊢ (∃!x ∈ A φ → ψ)
12	3, 11	jca 290	1 ⊢ (∃!x ∈ A φ → (B ∈ A ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  Ⅎwnf 1329   ∈ wcel 1374  ∃!wreu 2286  ℩crio 5392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794  df-riota 5393

This theorem is referenced by: (None)