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Theorem riotaclbgBAD 33258
 Description: Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotaclbgBAD (𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem riotaclbgBAD
StepHypRef Expression
1 riotacl 6525 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
2 undefnel2 7290 . . . 4 (𝐴𝑉 → ¬ (Undef‘𝐴) ∈ 𝐴)
3 iffalse 4045 . . . . . . 7 (¬ ∃!𝑥𝐴 𝜑 → if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴})) = (Undef‘{𝑥𝑥𝐴}))
4 ax-riotaBAD 33257 . . . . . . 7 (𝑥𝐴 𝜑) = if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴}))
5 abid1 2731 . . . . . . . 8 𝐴 = {𝑥𝑥𝐴}
65fveq2i 6106 . . . . . . 7 (Undef‘𝐴) = (Undef‘{𝑥𝑥𝐴})
73, 4, 63eqtr4g 2669 . . . . . 6 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = (Undef‘𝐴))
87eleq1d 2672 . . . . 5 (¬ ∃!𝑥𝐴 𝜑 → ((𝑥𝐴 𝜑) ∈ 𝐴 ↔ (Undef‘𝐴) ∈ 𝐴))
98notbid 307 . . . 4 (¬ ∃!𝑥𝐴 𝜑 → (¬ (𝑥𝐴 𝜑) ∈ 𝐴 ↔ ¬ (Undef‘𝐴) ∈ 𝐴))
102, 9syl5ibrcom 236 . . 3 (𝐴𝑉 → (¬ ∃!𝑥𝐴 𝜑 → ¬ (𝑥𝐴 𝜑) ∈ 𝐴))
1110con4d 113 . 2 (𝐴𝑉 → ((𝑥𝐴 𝜑) ∈ 𝐴 → ∃!𝑥𝐴 𝜑))
121, 11impbid2 215 1 (𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  {cab 2596  ∃!wreu 2898  ifcif 4036  ℩cio 5766  ‘cfv 5804  ℩crio 6510  Undefcund 7285 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-undef 7286 This theorem is referenced by:  riotaclbBAD  33259  riotasvd  33260
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