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Theorem riotass2 7418
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 4332 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)
2 simplr 769 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∀𝑥𝐴 (𝜑𝜓))
3 riotasbc 7406 . . . . 5 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 riotacl 7405 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
5 rspsbc 3888 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → (∀𝑥𝐴 (𝜑𝜓) → [(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓)))
6 sbcimg 3843 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → ([(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓) ↔ ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
75, 6sylibd 239 . . . . . 6 ((𝑥𝐴 𝜑) ∈ 𝐴 → (∀𝑥𝐴 (𝜑𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
84, 7syl 17 . . . . 5 (∃!𝑥𝐴 𝜑 → (∀𝑥𝐴 (𝜑𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
93, 8mpid 44 . . . 4 (∃!𝑥𝐴 𝜑 → (∀𝑥𝐴 (𝜑𝜓) → [(𝑥𝐴 𝜑) / 𝑥]𝜓))
101, 2, 9sylc 65 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → [(𝑥𝐴 𝜑) / 𝑥]𝜓)
111, 4syl 17 . . . . 5 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) ∈ 𝐴)
12 ssel 3989 . . . . . 6 (𝐴𝐵 → ((𝑥𝐴 𝜑) ∈ 𝐴 → (𝑥𝐴 𝜑) ∈ 𝐵))
1312ad2antrr 726 . . . . 5 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ((𝑥𝐴 𝜑) ∈ 𝐴 → (𝑥𝐴 𝜑) ∈ 𝐵))
1411, 13mpd 15 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) ∈ 𝐵)
15 simprr 773 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐵 𝜓)
16 nfriota1 7395 . . . . 5 𝑥(𝑥𝐴 𝜑)
1716nfsbc1 3810 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜓
18 sbceq1a 3802 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜓[(𝑥𝐴 𝜑) / 𝑥]𝜓))
1916, 17, 18riota2f 7412 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2014, 15, 19syl2anc 584 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2110, 20mpbid 232 . 2 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑))
2221eqcomd 2741 1 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  ∃!wreu 3376  [wsbc 3791  wss 3963  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by:  fisupcl  9507  quotlem  26357  adjbdln  32112  rexdiv  32893  cdlemefrs32fva  40383  addinvcom  42438
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