Theorem bnj1366 35379

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 1165	. . 3 ⊢ (𝜓 → 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
3	1	simp2bi 1164	. . . . 5 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑)
4		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
5		nfeu1 2614	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦𝜑
6	4, 5	nfralw 3309	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
7		nfra1 3286	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
8		rspa 3251	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦𝜑)
9		iota1 6507	. . . . . . . . . 10 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10		eqcom 2767	. . . . . . . . . 10 ⊢ ((℩𝑦𝜑) = 𝑦 ↔ 𝑦 = (℩𝑦𝜑))
11	9, 10	bitrdi 290	. . . . . . . . 9 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
12	8, 11	syl 18	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
13	7, 12	rexbida 3274	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑)))
14		abid 2742	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
15		eqid 2760	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) = (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
16		iotaex 6504	. . . . . . . 8 ⊢ (℩𝑦𝜑) ∈ V
17	15, 16	elrnmpti 5941	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑))
18	13, 14, 17	3bitr4g 317	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
19	6, 18	alrimi 2249	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
20	3, 19	syl 18	. . . 4 ⊢ (𝜓 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
21		nfab1 2924	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
22		nfiota1 6486	. . . . . . 7 ⊢ Ⅎ𝑦(℩𝑦𝜑)
23	4, 22	nfmpt 5203	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
24	23	nfrn 5931	. . . . 5 ⊢ Ⅎ𝑦ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
25	21, 24	cleqf 2950	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
26	20, 25	sylibr 237	. . 3 ⊢ (𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
27	2, 26	eqtrd 2795	. 2 ⊢ (𝜓 → 𝐵 = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
28	1	simp1bi 1163	. . 3 ⊢ (𝜓 → 𝐴 ∈ V)
29		mptexg 7216	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30		rnexg 7898	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
31	28, 29, 30	3syl 19	. 2 ⊢ (𝜓 → ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
32	27, 31	eqeltrd 2860	1 ⊢ (𝜓 → 𝐵 ∈ V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃!weu 2593  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ↦ cmpt 5186  ran crn 5649  ℩cio 6482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536

This theorem is used by:  bnj1489  35606