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Theorem bnj1366 30154
 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 1071 . . 3 (𝜓𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑})
31simp2bi 1070 . . . . 5 (𝜓 → ∀𝑥𝐴 ∃!𝑦𝜑)
4 nfcv 2751 . . . . . . 7 𝑦𝐴
5 nfeu1 2468 . . . . . . 7 𝑦∃!𝑦𝜑
64, 5nfral 2929 . . . . . 6 𝑦𝑥𝐴 ∃!𝑦𝜑
7 nfra1 2925 . . . . . . . 8 𝑥𝑥𝐴 ∃!𝑦𝜑
8 rspa 2914 . . . . . . . . 9 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → ∃!𝑦𝜑)
9 iota1 5782 . . . . . . . . . 10 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10 eqcom 2617 . . . . . . . . . 10 ((℩𝑦𝜑) = 𝑦𝑦 = (℩𝑦𝜑))
119, 10syl6bb 275 . . . . . . . . 9 (∃!𝑦𝜑 → (𝜑𝑦 = (℩𝑦𝜑)))
128, 11syl 17 . . . . . . . 8 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → (𝜑𝑦 = (℩𝑦𝜑)))
137, 12rexbida 3029 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝜑 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑)))
14 abid 2598 . . . . . . 7 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
15 eqid 2610 . . . . . . . 8 (𝑥𝐴 ↦ (℩𝑦𝜑)) = (𝑥𝐴 ↦ (℩𝑦𝜑))
16 iotaex 5785 . . . . . . . 8 (℩𝑦𝜑) ∈ V
1715, 16elrnmpti 5297 . . . . . . 7 (𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑))
1813, 14, 173bitr4g 302 . . . . . 6 (∀𝑥𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
196, 18alrimi 2069 . . . . 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 2753 . . . . 5 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
22 nfiota1 5770 . . . . . . 7 𝑦(℩𝑦𝜑)
234, 22nfmpt 4674 . . . . . 6 𝑦(𝑥𝐴 ↦ (℩𝑦𝜑))
2423nfrn 5289 . . . . 5 𝑦ran (𝑥𝐴 ↦ (℩𝑦𝜑))
2521, 24cleqf 2776 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
2620, 25sylibr 223 . . 3 (𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
272, 26eqtrd 2644 . 2 (𝜓𝐵 = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
281simp1bi 1069 . . 3 (𝜓𝐴 ∈ V)
29 mptexg 6389 . . 3 (𝐴 ∈ V → (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30 rnexg 6990 . . 3 ((𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3128, 29, 303syl 18 . 2 (𝜓 → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3227, 31eqeltrd 2688 1 (𝜓𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ↦ cmpt 4643  ran crn 5039  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  bnj1489  30378
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