Theorem riotaclbgBAD 40105

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 7395	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
2		undefnel2 8299	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ (Undef‘𝐴) ∈ 𝐴)
3		iffalse 4490	. . . . . . 7 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
4		ax-riotaBAD 40104	. . . . . . 7 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
5		abid1 2896	. . . . . . . 8 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
6	5	fveq2i 6889	. . . . . . 7 ⊢ (Undef‘𝐴) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})
7	3, 4, 6	3eqtr4g 2820	. . . . . 6 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = (Undef‘𝐴))
8	7	eleq1d 2845	. . . . 5 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ (Undef‘𝐴) ∈ 𝐴))
9	8	notbid 321	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ¬ (Undef‘𝐴) ∈ 𝐴))
10	2, 9	syl5ibrcom 250	. . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
11	10	con4d 116	. 2 ⊢ (𝐴 ∈ 𝑉 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → ∃!𝑥 ∈ 𝐴 𝜑))
12	1, 11	impbid2 229	1 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  {cab 2738  ∃!wreu 3363  ifcif 4481  ℩cio 6492  ‘cfv 6539  ℩crio 7377  Undefcund 8293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-riotaBAD 40104

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-riota 7378  df-undef 8294

This theorem is used by:  riotaclbBAD  40106  riotasvd  40107