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Theorem riotass2 5688
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
Assertion
Ref Expression
riotass2 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem riotass2
StepHypRef Expression
1 reuss2 3303 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)
2 simplr 500 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∀𝑥𝐴 (𝜑𝜓))
3 riotasbc 5677 . . . . 5 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 riotacl 5676 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
5 rspsbc 2943 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → (∀𝑥𝐴 (𝜑𝜓) → [(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓)))
6 sbcimg 2902 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → ([(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓) ↔ ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
75, 6sylibd 148 . . . . . 6 ((𝑥𝐴 𝜑) ∈ 𝐴 → (∀𝑥𝐴 (𝜑𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
84, 7syl 14 . . . . 5 (∃!𝑥𝐴 𝜑 → (∀𝑥𝐴 (𝜑𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
93, 8mpid 42 . . . 4 (∃!𝑥𝐴 𝜑 → (∀𝑥𝐴 (𝜑𝜓) → [(𝑥𝐴 𝜑) / 𝑥]𝜓))
101, 2, 9sylc 62 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → [(𝑥𝐴 𝜑) / 𝑥]𝜓)
111, 4syl 14 . . . . 5 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) ∈ 𝐴)
12 ssel 3041 . . . . . 6 (𝐴𝐵 → ((𝑥𝐴 𝜑) ∈ 𝐴 → (𝑥𝐴 𝜑) ∈ 𝐵))
1312ad2antrr 475 . . . . 5 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ((𝑥𝐴 𝜑) ∈ 𝐴 → (𝑥𝐴 𝜑) ∈ 𝐵))
1411, 13mpd 13 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) ∈ 𝐵)
15 simprr 502 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐵 𝜓)
16 nfriota1 5669 . . . . 5 𝑥(𝑥𝐴 𝜑)
1716nfsbc1 2879 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜓
18 sbceq1a 2871 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜓[(𝑥𝐴 𝜑) / 𝑥]𝜓))
1916, 17, 18riota2f 5683 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2014, 15, 19syl2anc 406 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2110, 20mpbid 146 . 2 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑))
2221eqcomd 2105 1 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1299  wcel 1448  wral 2375  wrex 2376  ∃!wreu 2377  [wsbc 2862  wss 3021  crio 5661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-uni 3684  df-iota 5024  df-riota 5662
This theorem is referenced by: (None)
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