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Theorem riotass 5801
 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 3388 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
2 riotasbc 5789 . . . 4 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
31, 2syl 14 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → [(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 simp1 982 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → 𝐴𝐵)
5 riotacl 5788 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
61, 5syl 14 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐴)
74, 6sseldd 3129 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐵)
8 simp3 984 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐵 𝜑)
9 nfriota1 5781 . . . . 5 𝑥(𝑥𝐴 𝜑)
109nfsbc1 2954 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜑
11 sbceq1a 2946 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑))
129, 10, 11riota2f 5795 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
137, 8, 12syl2anc 409 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
143, 13mpbid 146 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑))
1514eqcomd 2163 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ∃wrex 2436  ∃!wreu 2437  [wsbc 2937   ⊆ wss 3102  ℩crio 5773 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-uni 3773  df-iota 5132  df-riota 5774 This theorem is referenced by:  moriotass  5802
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