Theorem riotass 5927

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotass	⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotass

Step	Hyp	Ref	Expression
1		reuss 3454	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑)
2		riotasbc 5915	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
3	1, 2	syl 14	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
4		simp1 1000	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → 𝐴 ⊆ 𝐵)
5		riotacl 5914	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
6	1, 5	syl 14	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
7	4, 6	sseldd 3194	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵)
8		simp3 1002	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐵 𝜑)
9		nfriota1 5907	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
10	9	nfsbc1 3016	. . . . 5 ⊢ Ⅎ𝑥[(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑
11		sbceq1a 3008	. . . . 5 ⊢ (𝑥 = (℩𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑))
12	9, 10, 11	riota2f 5921	. . . 4 ⊢ (((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑)))
13	7, 8, 12	syl2anc 411	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑)))
14	3, 13	mpbid 147	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑))
15	14	eqcomd 2211	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  ∃wrex 2485  ∃!wreu 2486  [wsbc 2998   ⊆ wss 3166  ℩crio 5898

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5232  df-riota 5899

This theorem is referenced by:  moriotass  5928