Theorem riota5f 5849

Description: A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota5f.1	⊢ (𝜑 → Ⅎ𝑥𝐵)
riota5f.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
riota5f.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))

Assertion

Ref	Expression
riota5f	⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

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

Proof of Theorem riota5f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		riota5f.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))
2	1	ralrimiva 2550	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵))
3		riota5f.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4		a1tru 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → ⊤)
5		reu6i 2928	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)) → ∃!𝑥 ∈ 𝐴 𝜓)
6	5	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → ∃!𝑥 ∈ 𝐴 𝜓)
7		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
8		nfv 1528	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
9		nfra1 2508	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)
10	8, 9	nfan 1565	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))
11	7, 10	nfan 1565	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)))
12		nfcvd 2320	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → Ⅎ𝑥𝑦)
13		nfvd 1529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → Ⅎ𝑥⊤)
14		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → 𝑦 ∈ 𝐴)
15		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
16		simplrr 536	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))
17		simplrl 535	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
18	15, 17	eqeltrd 2254	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝐴)
19		rsp 2524	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝑥 = 𝑦)))
20	16, 18, 19	sylc 62	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝑥 = 𝑦))
21	15, 20	mpbird 167	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝜓)
22		a1tru 1369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ⊤)
23	21, 22	2thd 175	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ ⊤))
24	11, 12, 13, 14, 23	riota2df 5845	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (⊤ ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
25	6, 24	mpdan 421	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → (⊤ ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
26	4, 25	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦)
27	26	expr 375	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
28	27	ralrimiva 2550	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
29		rspsbc 3045	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦) → [𝐵 / 𝑦](∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦)))
30	3, 28, 29	sylc 62	. . 3 ⊢ (𝜑 → [𝐵 / 𝑦](∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
31		nfcvd 2320	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑦)
32		riota5f.1	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐵)
33	31, 32	nfeqd 2334	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝐵)
34	7, 33	nfan1 1564	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 = 𝐵)
35		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
36	35	eqeq2d 2189	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵))
37	36	bibi2d 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐵)))
38	34, 37	ralbid 2475	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵)))
39	35	eqeq2d 2189	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((℩𝑥 ∈ 𝐴 𝜓) = 𝑦 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
40	38, 39	imbi12d 234	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦) ↔ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)))
41	3, 40	sbcied 2999	. . 3 ⊢ (𝜑 → ([𝐵 / 𝑦](∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦) ↔ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)))
42	30, 41	mpbid 147	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
43	2, 42	mpd 13	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ⊤wtru 1354   ∈ wcel 2148  Ⅎwnfc 2306  ∀wral 2455  ∃!wreu 2457  [wsbc 2962  ℩crio 5824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3597  df-pr 3598  df-uni 3808  df-iota 5174  df-riota 5825

This theorem is referenced by:  riota5  5850