Theorem riotaxfrd 7205

Description: Change the variable 𝑥 in the expression for "the unique 𝑥 such that 𝜓 " to another variable 𝑦 contained in expression 𝐵. Use reuhypd 5312 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riotaxfrd.1	⊢ Ⅎ𝑦𝐶
riotaxfrd.2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ 𝐴)
riotaxfrd.3	⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → 𝐶 ∈ 𝐴)
riotaxfrd.4	⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
riotaxfrd.5	⊢ (𝑦 = (℩𝑦 ∈ 𝐴 𝜒) → 𝐵 = 𝐶)
riotaxfrd.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐴 𝑥 = 𝐵)

Assertion

Ref	Expression
riotaxfrd	⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐶)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑦,𝐴   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem riotaxfrd

Step	Hyp	Ref	Expression
1		rabid 3290	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
2	1	baib 539	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ 𝜓))
3	2	riotabiia 7191	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = (℩𝑥 ∈ 𝐴 𝜓)
4		riotaxfrd.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ 𝐴)
5		riotaxfrd.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐴 𝑥 = 𝐵)
6		riotaxfrd.4	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
7	4, 5, 6	reuxfr1ds 3664	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))
8		riotacl2 7187	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 → (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒})
9	8	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒})
10		riotacl 7188	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 → (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴)
11		nfriota1 7177	. . . . . . . . 9 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐴 𝜒)
12		riotaxfrd.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐶
13		riotaxfrd.5	. . . . . . . . 9 ⊢ (𝑦 = (℩𝑦 ∈ 𝐴 𝜒) → 𝐵 = 𝐶)
14	11, 12, 4, 6, 13	rabxfrd 5310	. . . . . . . 8 ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒}))
15	10, 14	sylan2 596	. . . . . . 7 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒}))
16	9, 15	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
17	16	ex 416	. . . . 5 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐴 𝜒 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
18	7, 17	sylbid 243	. . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
19	18	imp 410	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
20		riotaxfrd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → 𝐶 ∈ 𝐴)
21	20	ex 416	. . . . . . 7 ⊢ (𝜑 → ((℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴 → 𝐶 ∈ 𝐴))
22	10, 21	syl5 34	. . . . . 6 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐴 𝜒 → 𝐶 ∈ 𝐴))
23	7, 22	sylbid 243	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 → 𝐶 ∈ 𝐴))
24	23	imp 410	. . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐶 ∈ 𝐴)
25	1	baibr 540	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
26	25	reubiia 3302	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
27	26	biimpi 219	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
28	27	adantl 485	. . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
29		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝐶
30		nfrab1 3296	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓}
31	30	nfel2 2922	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}
32		eleq1 2825	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
33	29, 31, 32	riota2f 7195	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶))
34	24, 28, 33	syl2anc 587	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶))
35	19, 34	mpbid 235	. 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶)
36	3, 35	eqtr3id 2792	1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Ⅎwnfc 2884  ∃!wreu 3063  {crab 3065  ℩crio 7169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-un 3871  df-in 3873  df-ss 3883  df-sn 4542  df-pr 4544  df-uni 4820  df-iota 6338  df-riota 7170

This theorem is referenced by:  riotaneg  11811  zriotaneg  12291  riotaocN  36960