Theorem moriotass 5984

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
moriotass	⊢ ((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 moriotass

Step	Hyp	Ref	Expression
1		ssrexv 3289	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))
2	1	imp 124	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐵 𝜑)
3	2	3adant3 1041	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) → ∃𝑥 ∈ 𝐵 𝜑)
4		simp3 1023	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) → ∃*𝑥 ∈ 𝐵 𝜑)
5		reu5 2749	. . 3 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))
6	3, 4, 5	sylanbrc 417	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐵 𝜑)
7		riotass 5983	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))
8	6, 7	syld3an3 1316	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∧ w3a 1002   = wceq 1395  ∃wrex 2509  ∃!wreu 2510  ∃*wrmo 2511   ⊆ wss 3197  ℩crio 5952

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3888  df-iota 5277  df-riota 5953

This theorem is referenced by: (None)