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Theorem riotass 5857
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 3416 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
2 riotasbc 5845 . . . 4 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
31, 2syl 14 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → [(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 simp1 997 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → 𝐴𝐵)
5 riotacl 5844 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
61, 5syl 14 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐴)
74, 6sseldd 3156 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐵)
8 simp3 999 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐵 𝜑)
9 nfriota1 5837 . . . . 5 𝑥(𝑥𝐴 𝜑)
109nfsbc1 2980 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜑
11 sbceq1a 2972 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑))
129, 10, 11riota2f 5851 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
137, 8, 12syl2anc 411 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
143, 13mpbid 147 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑))
1514eqcomd 2183 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  w3a 978   = wceq 1353  wcel 2148  wrex 2456  ∃!wreu 2457  [wsbc 2962  wss 3129  crio 5829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810  df-iota 5178  df-riota 5830
This theorem is referenced by:  moriotass  5858
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