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Theorem riotaclbgBAD 38350
Description: Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotaclbgBAD (𝐴 ∈ 𝑉 β†’ (βˆƒ!π‘₯ ∈ 𝐴 πœ‘ ↔ (β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴))
Distinct variable group:   π‘₯,𝐴
Allowed substitution hints:   πœ‘(π‘₯)   𝑉(π‘₯)

Proof of Theorem riotaclbgBAD
StepHypRef Expression
1 riotacl 7388 . 2 (βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ (β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴)
2 undefnel2 8274 . . . 4 (𝐴 ∈ 𝑉 β†’ Β¬ (Undefβ€˜π΄) ∈ 𝐴)
3 iffalse 4533 . . . . . . 7 (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ if(βˆƒ!π‘₯ ∈ 𝐴 πœ‘, (β„©π‘₯(π‘₯ ∈ 𝐴 ∧ πœ‘)), (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴})) = (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴}))
4 ax-riotaBAD 38349 . . . . . . 7 (β„©π‘₯ ∈ 𝐴 πœ‘) = if(βˆƒ!π‘₯ ∈ 𝐴 πœ‘, (β„©π‘₯(π‘₯ ∈ 𝐴 ∧ πœ‘)), (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴}))
5 abid1 2865 . . . . . . . 8 𝐴 = {π‘₯ ∣ π‘₯ ∈ 𝐴}
65fveq2i 6894 . . . . . . 7 (Undefβ€˜π΄) = (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴})
73, 4, 63eqtr4g 2792 . . . . . 6 (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ (β„©π‘₯ ∈ 𝐴 πœ‘) = (Undefβ€˜π΄))
87eleq1d 2813 . . . . 5 (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ ((β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴 ↔ (Undefβ€˜π΄) ∈ 𝐴))
98notbid 318 . . . 4 (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ (Β¬ (β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴 ↔ Β¬ (Undefβ€˜π΄) ∈ 𝐴))
102, 9syl5ibrcom 246 . . 3 (𝐴 ∈ 𝑉 β†’ (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ Β¬ (β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴))
1110con4d 115 . 2 (𝐴 ∈ 𝑉 β†’ ((β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴 β†’ βˆƒ!π‘₯ ∈ 𝐴 πœ‘))
121, 11impbid2 225 1 (𝐴 ∈ 𝑉 β†’ (βˆƒ!π‘₯ ∈ 𝐴 πœ‘ ↔ (β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  {cab 2704  βˆƒ!wreu 3369  ifcif 4524  β„©cio 6492  β€˜cfv 6542  β„©crio 7369  Undefcund 8269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-riotaBAD 38349
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7370  df-undef 8270
This theorem is referenced by:  riotaclbBAD  38351  riotasvd  38352
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