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Theorem riotass 6679
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotass ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotass
StepHypRef Expression
1 reuss 3941 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
2 riotasbc 6666 . . . 4 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
31, 2syl 17 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → [(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 simp1 1081 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → 𝐴𝐵)
5 riotacl 6665 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
61, 5syl 17 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐴)
74, 6sseldd 3637 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐵)
8 simp3 1083 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐵 𝜑)
9 nfriota1 6658 . . . . 5 𝑥(𝑥𝐴 𝜑)
109nfsbc1 3487 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜑
11 sbceq1a 3479 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑))
129, 10, 11riota2f 6672 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
137, 8, 12syl2anc 694 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
143, 13mpbid 222 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑))
1514eqcomd 2657 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054   = wceq 1523  wcel 2030  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:  moriotass  6680
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