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Theorem riotaclbgBAD 37445
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 7336 . 2 (βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ (β„©π‘₯ ∈ 𝐴 πœ‘) ∈ 𝐴)
2 undefnel2 8213 . . . 4 (𝐴 ∈ 𝑉 β†’ Β¬ (Undefβ€˜π΄) ∈ 𝐴)
3 iffalse 4500 . . . . . . 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 37444 . . . . . . 7 (β„©π‘₯ ∈ 𝐴 πœ‘) = if(βˆƒ!π‘₯ ∈ 𝐴 πœ‘, (β„©π‘₯(π‘₯ ∈ 𝐴 ∧ πœ‘)), (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴}))
5 abid1 2875 . . . . . . . 8 𝐴 = {π‘₯ ∣ π‘₯ ∈ 𝐴}
65fveq2i 6850 . . . . . . 7 (Undefβ€˜π΄) = (Undefβ€˜{π‘₯ ∣ π‘₯ ∈ 𝐴})
73, 4, 63eqtr4g 2802 . . . . . 6 (Β¬ βˆƒ!π‘₯ ∈ 𝐴 πœ‘ β†’ (β„©π‘₯ ∈ 𝐴 πœ‘) = (Undefβ€˜π΄))
87eleq1d 2823 . . . . 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 247 . . 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 397   ∈ wcel 2107  {cab 2714  βˆƒ!wreu 3354  ifcif 4491  β„©cio 6451  β€˜cfv 6501  β„©crio 7317  Undefcund 8208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-riotaBAD 37444
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-riota 7318  df-undef 8209
This theorem is referenced by:  riotaclbBAD  37446  riotasvd  37447
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