Theorem bnj1366 32809

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1366.1	⊢ (𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}))

Assertion

Ref	Expression
bnj1366	⊢ (𝜓 → 𝐵 ∈ V)

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem bnj1366

Step	Hyp	Ref	Expression
1		bnj1366.1	. . . 4 ⊢ (𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}))
2	1	simp3bi 1146	. . 3 ⊢ (𝜓 → 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
3	1	simp2bi 1145	. . . . 5 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑)
4		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
5		nfeu1 2588	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦𝜑
6	4, 5	nfralw 3151	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
7		nfra1 3144	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
8		rspa 3132	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦𝜑)
9		iota1 6410	. . . . . . . . . 10 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10		eqcom 2745	. . . . . . . . . 10 ⊢ ((℩𝑦𝜑) = 𝑦 ↔ 𝑦 = (℩𝑦𝜑))
11	9, 10	bitrdi 287	. . . . . . . . 9 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
12	8, 11	syl 17	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
13	7, 12	rexbida 3251	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑)))
14		abid 2719	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
15		eqid 2738	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) = (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
16		iotaex 6413	. . . . . . . 8 ⊢ (℩𝑦𝜑) ∈ V
17	15, 16	elrnmpti 5869	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑))
18	13, 14, 17	3bitr4g 314	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
19	6, 18	alrimi 2206	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
20	3, 19	syl 17	. . . 4 ⊢ (𝜓 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
21		nfab1 2909	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
22		nfiota1 6393	. . . . . . 7 ⊢ Ⅎ𝑦(℩𝑦𝜑)
23	4, 22	nfmpt 5181	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
24	23	nfrn 5861	. . . . 5 ⊢ Ⅎ𝑦ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
25	21, 24	cleqf 2938	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
26	20, 25	sylibr 233	. . 3 ⊢ (𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
27	2, 26	eqtrd 2778	. 2 ⊢ (𝜓 → 𝐵 = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
28	1	simp1bi 1144	. . 3 ⊢ (𝜓 → 𝐴 ∈ V)
29		mptexg 7097	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30		rnexg 7751	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
31	28, 29, 30	3syl 18	. 2 ⊢ (𝜓 → ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
32	27, 31	eqeltrd 2839	1 ⊢ (𝜓 → 𝐵 ∈ V)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  {cab 2715  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ↦ cmpt 5157  ran crn 5590  ℩cio 6389

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by:  bnj1489  33036