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Theorem riotass 7224
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 4248 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
2 riotasbc 7211 . . . 4 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
31, 2syl 17 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → [(𝑥𝐴 𝜑) / 𝑥]𝜑)
4 simp1 1138 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → 𝐴𝐵)
5 riotacl 7210 . . . . . 6 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
61, 5syl 17 . . . . 5 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐴)
74, 6sseldd 3919 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) ∈ 𝐵)
8 simp3 1140 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐵 𝜑)
9 nfriota1 7199 . . . . 5 𝑥(𝑥𝐴 𝜑)
109nfsbc1 3730 . . . . 5 𝑥[(𝑥𝐴 𝜑) / 𝑥]𝜑
11 sbceq1a 3722 . . . . 5 (𝑥 = (𝑥𝐴 𝜑) → (𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑))
129, 10, 11riota2f 7217 . . . 4 (((𝑥𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
137, 8, 12syl2anc 587 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑)))
143, 13mpbid 235 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = (𝑥𝐴 𝜑))
1514eqcomd 2745 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1089   = wceq 1543  wcel 2112  wrex 3065  ∃!wreu 3066  [wsbc 3711  wss 3883  crio 7191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6359  df-riota 7192
This theorem is referenced by:  moriotass  7225  resubeqsub  40167
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