Theorem bnj1366 32211

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 1144	. . 3 ⊢ (𝜓 → 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
3	1	simp2bi 1143	. . . . 5 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑)
4		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
5		nfeu1 2649	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦𝜑
6	4, 5	nfralw 3189	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
7		nfra1 3183	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
8		rspa 3171	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦𝜑)
9		iota1 6301	. . . . . . . . . 10 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10		eqcom 2805	. . . . . . . . . 10 ⊢ ((℩𝑦𝜑) = 𝑦 ↔ 𝑦 = (℩𝑦𝜑))
11	9, 10	syl6bb 290	. . . . . . . . 9 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
12	8, 11	syl 17	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
13	7, 12	rexbida 3277	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑)))
14		abid 2780	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
15		eqid 2798	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) = (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
16		iotaex 6304	. . . . . . . 8 ⊢ (℩𝑦𝜑) ∈ V
17	15, 16	elrnmpti 5796	. . . . . . 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 2211	. . . . 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 2957	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
22		nfiota1 6285	. . . . . . 7 ⊢ Ⅎ𝑦(℩𝑦𝜑)
23	4, 22	nfmpt 5127	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
24	23	nfrn 5788	. . . . 5 ⊢ Ⅎ𝑦ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
25	21, 24	cleqf 2983	. . . 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 2833	. 2 ⊢ (𝜓 → 𝐵 = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
28	1	simp1bi 1142	. . 3 ⊢ (𝜓 → 𝐴 ∈ V)
29		mptexg 6961	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30		rnexg 7595	. . 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 2890	1 ⊢ (𝜓 → 𝐵 ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃!weu 2628  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ↦ cmpt 5110  ran crn 5520  ℩cio 6281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by:  bnj1489  32438