Theorem riotaclbgBAD 39461

Description: Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotaclbgBAD	⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem riotaclbgBAD

Step	Hyp	Ref	Expression
1		riotacl 7334	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
2		undefnel2 8221	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ (Undef‘𝐴) ∈ 𝐴)
3		iffalse 4466	. . . . . . 7 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
4		ax-riotaBAD 39460	. . . . . . 7 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
5		abid1 2877	. . . . . . . 8 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
6	5	fveq2i 6834	. . . . . . 7 ⊢ (Undef‘𝐴) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})
7	3, 4, 6	3eqtr4g 2801	. . . . . 6 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = (Undef‘𝐴))
8	7	eleq1d 2826	. . . . 5 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ (Undef‘𝐴) ∈ 𝐴))
9	8	notbid 320	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ¬ (Undef‘𝐴) ∈ 𝐴))
10	2, 9	syl5ibrcom 249	. . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
11	10	con4d 115	. 2 ⊢ (𝐴 ∈ 𝑉 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → ∃!𝑥 ∈ 𝐴 𝜑))
12	1, 11	impbid2 228	1 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  {cab 2719  ∃!wreu 3344  ifcif 4457  ℩cio 6443  ‘cfv 6489  ℩crio 7316  Undefcund 8216

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-riotaBAD 39460

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-riota 7317  df-undef 8217

This theorem is referenced by:  riotaclbBAD  39462  riotasvd  39463