Theorem bnj1366 35126

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 1161	. . 3 ⊢ (𝜓 → 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
3	1	simp2bi 1160	. . . . 5 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑)
4		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
5		nfeu1 2618	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦𝜑
6	4, 5	nfralw 3311	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
7		nfra1 3288	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
8		rspa 3253	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦𝜑)
9		iota1 6502	. . . . . . . . . 10 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10		eqcom 2771	. . . . . . . . . 10 ⊢ ((℩𝑦𝜑) = 𝑦 ↔ 𝑦 = (℩𝑦𝜑))
11	9, 10	bitrdi 289	. . . . . . . . 9 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
12	8, 11	syl 17	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝑦 = (℩𝑦𝜑)))
13	7, 12	rexbida 3276	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑)))
14		abid 2746	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
15		eqid 2764	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) = (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
16		iotaex 6499	. . . . . . . 8 ⊢ (℩𝑦𝜑) ∈ V
17	15, 16	elrnmpti 5940	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑))
18	13, 14, 17	3bitr4g 316	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
19	6, 18	alrimi 2250	. . . . 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 2928	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
22		nfiota1 6481	. . . . . . 7 ⊢ Ⅎ𝑦(℩𝑦𝜑)
23	4, 22	nfmpt 5200	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
24	23	nfrn 5930	. . . . 5 ⊢ Ⅎ𝑦ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))
25	21, 24	cleqf 2954	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))))
26	20, 25	sylibr 236	. . 3 ⊢ (𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
27	2, 26	eqtrd 2799	. 2 ⊢ (𝜓 → 𝐵 = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))
28	1	simp1bi 1159	. . 3 ⊢ (𝜓 → 𝐴 ∈ V)
29		mptexg 7207	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30		rnexg 7885	. . 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 2864	1 ⊢ (𝜓 → 𝐵 ∈ V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099  ∀wal 1560   = wceq 1562   ∈ wcel 2144  ∃!weu 2597  {cab 2742  ∀wral 3078  ∃wrex 3088  Vcvv 3456   ↦ cmpt 5183  ran crn 5650  ℩cio 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531

This theorem is referenced by:  bnj1489  35353