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Theorem bnj1366 31026
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 ∧ ∀𝑥𝐴 ∃!𝑦𝜑𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑}))
Assertion
Ref Expression
bnj1366 (𝜓𝐵 ∈ V)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem bnj1366
StepHypRef Expression
1 bnj1366.1 . . . 4 (𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥𝐴 ∃!𝑦𝜑𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑}))
21simp3bi 1098 . . 3 (𝜓𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑})
31simp2bi 1097 . . . . 5 (𝜓 → ∀𝑥𝐴 ∃!𝑦𝜑)
4 nfcv 2793 . . . . . . 7 𝑦𝐴
5 nfeu1 2508 . . . . . . 7 𝑦∃!𝑦𝜑
64, 5nfral 2974 . . . . . 6 𝑦𝑥𝐴 ∃!𝑦𝜑
7 nfra1 2970 . . . . . . . 8 𝑥𝑥𝐴 ∃!𝑦𝜑
8 rspa 2959 . . . . . . . . 9 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → ∃!𝑦𝜑)
9 iota1 5903 . . . . . . . . . 10 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10 eqcom 2658 . . . . . . . . . 10 ((℩𝑦𝜑) = 𝑦𝑦 = (℩𝑦𝜑))
119, 10syl6bb 276 . . . . . . . . 9 (∃!𝑦𝜑 → (𝜑𝑦 = (℩𝑦𝜑)))
128, 11syl 17 . . . . . . . 8 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → (𝜑𝑦 = (℩𝑦𝜑)))
137, 12rexbida 3076 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝜑 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑)))
14 abid 2639 . . . . . . 7 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
15 eqid 2651 . . . . . . . 8 (𝑥𝐴 ↦ (℩𝑦𝜑)) = (𝑥𝐴 ↦ (℩𝑦𝜑))
16 iotaex 5906 . . . . . . . 8 (℩𝑦𝜑) ∈ V
1715, 16elrnmpti 5408 . . . . . . 7 (𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑))
1813, 14, 173bitr4g 303 . . . . . 6 (∀𝑥𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
196, 18alrimi 2120 . . . . 5 (∀𝑥𝐴 ∃!𝑦𝜑 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
203, 19syl 17 . . . 4 (𝜓 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
21 nfab1 2795 . . . . 5 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
22 nfiota1 5891 . . . . . . 7 𝑦(℩𝑦𝜑)
234, 22nfmpt 4779 . . . . . 6 𝑦(𝑥𝐴 ↦ (℩𝑦𝜑))
2423nfrn 5400 . . . . 5 𝑦ran (𝑥𝐴 ↦ (℩𝑦𝜑))
2521, 24cleqf 2819 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
2620, 25sylibr 224 . . 3 (𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
272, 26eqtrd 2685 . 2 (𝜓𝐵 = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
281simp1bi 1096 . . 3 (𝜓𝐴 ∈ V)
29 mptexg 6525 . . 3 (𝐴 ∈ V → (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30 rnexg 7140 . . 3 ((𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3128, 29, 303syl 18 . 2 (𝜓 → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3227, 31eqeltrd 2730 1 (𝜓𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  ∃!weu 2498  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  cmpt 4762  ran crn 5144  cio 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  bnj1489  31250
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