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Theorem riotaclbgBAD 38478
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 7387 . 2 (βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ (β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴)
2 undefnel2 8276 . . . 4 (𝐴 ∈ 𝑉 β†’ Β¬ (Undefβ€˜π΄) ∈ 𝐴)
3 iffalse 4534 . . . . . . 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 38477 . . . . . . 7 (β„©π‘₯ ∈ 𝐴 πœ‘) = if(βˆƒ!π‘₯ ∈ 𝐴 πœ‘, (β„©π‘₯(π‘₯ ∈ 𝐴 ∧ πœ‘)), (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴}))
5 abid1 2862 . . . . . . . 8 𝐴 = {π‘₯ ∣ π‘₯ ∈ 𝐴}
65fveq2i 6893 . . . . . . 7 (Undefβ€˜π΄) = (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴})
73, 4, 63eqtr4g 2790 . . . . . 6 (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ (β„©π‘₯ ∈ 𝐴 πœ‘) = (Undefβ€˜π΄))
87eleq1d 2810 . . . . 5 (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ ((β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴 ↔ (Undefβ€˜π΄) ∈ 𝐴))
98notbid 317 . . . 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 394   ∈ wcel 2098  {cab 2702  βˆƒ!wreu 3362  ifcif 4525  β„©cio 6493  β€˜cfv 6543  β„©crio 7368  Undefcund 8271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-riotaBAD 38477
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-riota 7369  df-undef 8272
This theorem is referenced by:  riotaclbBAD  38479  riotasvd  38480
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