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Theorem riotass2 6678
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 3940 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)
2 simplr 807 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∀𝑥𝐴 (𝜑𝜓))
3 riotasbc 6666 . . . . 5 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 riotacl 6665 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
5 rspsbc 3551 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → (∀𝑥𝐴 (𝜑𝜓) → [(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓)))
6 sbcimg 3510 . . . . . . 7 ((𝑥𝐴 𝜑) ∈ 𝐴 → ([(𝑥𝐴 𝜑) / 𝑥](𝜑𝜓) ↔ ([(𝑥𝐴 𝜑) / 𝑥]𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜓)))
75, 6sylibd 229 . . . . . 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 3630 . . . . . 6 (𝐴𝐵 → ((𝑥𝐴 𝜑) ∈ 𝐴 → (𝑥𝐴 𝜑) ∈ 𝐵))
1312ad2antrr 762 . . . . 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 811 . . . 4 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐵 𝜓)
16 nfriota1 6658 . . . . 5 𝑥(𝑥𝐴 𝜑)
1716nfsbc1 3487 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜓
18 sbceq1a 3479 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜓[(𝑥𝐴 𝜑) / 𝑥]𝜓))
1916, 17, 18riota2f 6672 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜓) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2014, 15, 19syl2anc 694 . . 3 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ([(𝑥𝐴 𝜑) / 𝑥]𝜓 ↔ (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑)))
2110, 20mpbid 222 . 2 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐵 𝜓) = (𝑥𝐴 𝜑))
2221eqcomd 2657 1 (((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  [wsbc 3468  wss 3607  crio 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889  df-riota 6651
This theorem is referenced by:  fisupcl  8416  quotlem  24100  adjbdln  29070  rexdiv  29762  cdlemefrs32fva  36005
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