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Theorem riotass2 7345
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 4276 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)
2 simplr 768 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∀𝑥𝐴 (𝜑𝜓))
3 riotasbc 7333 . . . . 5 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 riotacl 7332 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
5 rspsbc 3836 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → (∀𝑥𝐴 (𝜑𝜓) → [(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓)))
6 sbcimg 3791 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → ([(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓) ↔ ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
75, 6sylibd 238 . . . . . 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 3938 . . . . . 6 (𝐴𝐵 → ((𝑥𝐴 𝜑) ∈ 𝐴 → (𝑥𝐴 𝜑) ∈ 𝐵))
1312ad2antrr 725 . . . . 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 772 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐵 𝜓)
16 nfriota1 7321 . . . . 5 𝑥(𝑥𝐴 𝜑)
1716nfsbc1 3759 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜓
18 sbceq1a 3751 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜓[(𝑥𝐴 𝜑) / 𝑥]𝜓))
1916, 17, 18riota2f 7339 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2014, 15, 19syl2anc 585 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2110, 20mpbid 231 . 2 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑))
2221eqcomd 2743 1 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  ∃!wreu 3352  [wsbc 3740  wss 3911  crio 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-un 3916  df-in 3918  df-ss 3928  df-sn 4588  df-pr 4590  df-uni 4867  df-iota 6449  df-riota 7314
This theorem is referenced by:  fisupcl  9406  quotlem  25663  adjbdln  31028  rexdiv  31785  cdlemefrs32fva  38866  addinvcom  40903
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