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Theorem riotass 7419
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 4333 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
2 riotasbc 7406 . . . 4 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
31, 2syl 17 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → [(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 simp1 1135 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → 𝐴𝐵)
5 riotacl 7405 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
61, 5syl 17 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐴)
74, 6sseldd 3996 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐵)
8 simp3 1137 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐵 𝜑)
9 nfriota1 7395 . . . . 5 𝑥(𝑥𝐴 𝜑)
109nfsbc1 3810 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜑
11 sbceq1a 3802 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑))
129, 10, 11riota2f 7412 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
137, 8, 12syl2anc 584 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
143, 13mpbid 232 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑))
1514eqcomd 2741 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  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:  moriotass  7420  resubeqsub  42436
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