Theorem bnj1366 34841

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 1147	. . 3 ⊢ (𝜓 → 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
3	1	simp2bi 1146	. . . . 5 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑)
4		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
5		nfeu1 2583	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦𝜑
6	4, 5	nfralw 3279	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
7		nfra1 3256	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
8		rspa 3221	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦𝜑)
9		iota1 6460	. . . . . . . . . 10 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10		eqcom 2738	. . . . . . . . . 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 3244	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑)))
14		abid 2713	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
15		eqid 2731	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) = (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
16		iotaex 6457	. . . . . . . 8 ⊢ (℩𝑦𝜑) ∈ V
17	15, 16	elrnmpti 5901	. . . . . . 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 2216	. . . . 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 2896	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
22		nfiota1 6439	. . . . . . 7 ⊢ Ⅎ𝑦(℩𝑦𝜑)
23	4, 22	nfmpt 5187	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
24	23	nfrn 5891	. . . . 5 ⊢ Ⅎ𝑦ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
25	21, 24	cleqf 2923	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
26	20, 25	sylibr 234	. . 3 ⊢ (𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
27	2, 26	eqtrd 2766	. 2 ⊢ (𝜓 → 𝐵 = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
28	1	simp1bi 1145	. . 3 ⊢ (𝜓 → 𝐴 ∈ V)
29		mptexg 7155	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30		rnexg 7832	. . 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 2831	1 ⊢ (𝜓 → 𝐵 ∈ V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ↦ cmpt 5170  ran crn 5615  ℩cio 6435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489

This theorem is referenced by:  bnj1489  35068