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Theorem bnj1366 34822
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 1146 . . 3 (𝜓𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑})
31simp2bi 1145 . . . . 5 (𝜓 → ∀𝑥𝐴 ∃!𝑦𝜑)
4 nfcv 2903 . . . . . . 7 𝑦𝐴
5 nfeu1 2586 . . . . . . 7 𝑦∃!𝑦𝜑
64, 5nfralw 3309 . . . . . 6 𝑦𝑥𝐴 ∃!𝑦𝜑
7 nfra1 3282 . . . . . . . 8 𝑥𝑥𝐴 ∃!𝑦𝜑
8 rspa 3246 . . . . . . . . 9 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → ∃!𝑦𝜑)
9 iota1 6540 . . . . . . . . . 10 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10 eqcom 2742 . . . . . . . . . 10 ((℩𝑦𝜑) = 𝑦𝑦 = (℩𝑦𝜑))
119, 10bitrdi 287 . . . . . . . . 9 (∃!𝑦𝜑 → (𝜑𝑦 = (℩𝑦𝜑)))
128, 11syl 17 . . . . . . . 8 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → (𝜑𝑦 = (℩𝑦𝜑)))
137, 12rexbida 3270 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝜑 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑)))
14 abid 2716 . . . . . . 7 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
15 eqid 2735 . . . . . . . 8 (𝑥𝐴 ↦ (℩𝑦𝜑)) = (𝑥𝐴 ↦ (℩𝑦𝜑))
16 iotaex 6536 . . . . . . . 8 (℩𝑦𝜑) ∈ V
1715, 16elrnmpti 5976 . . . . . . 7 (𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑))
1813, 14, 173bitr4g 314 . . . . . 6 (∀𝑥𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
196, 18alrimi 2211 . . . . 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 2905 . . . . 5 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
22 nfiota1 6518 . . . . . . 7 𝑦(℩𝑦𝜑)
234, 22nfmpt 5255 . . . . . 6 𝑦(𝑥𝐴 ↦ (℩𝑦𝜑))
2423nfrn 5966 . . . . 5 𝑦ran (𝑥𝐴 ↦ (℩𝑦𝜑))
2521, 24cleqf 2932 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
2620, 25sylibr 234 . . 3 (𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
272, 26eqtrd 2775 . 2 (𝜓𝐵 = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
281simp1bi 1144 . . 3 (𝜓𝐴 ∈ V)
29 mptexg 7241 . . 3 (𝐴 ∈ V → (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30 rnexg 7925 . . 3 ((𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3128, 29, 303syl 18 . 2 (𝜓 → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3227, 31eqeltrd 2839 1 (𝜓𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  ∃!weu 2566  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cmpt 5231  ran crn 5690  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  bnj1489  35049
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