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Theorem iinabrex 32589
Description: Rewriting an indexed intersection into an intersection of its image set. (Contributed by Thierry Arnoux, 15-Jun-2024.)
Assertion
Ref Expression
iinabrex (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem iinabrex
Dummy variables 𝑡 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfra1 3282 . . . . . . . 8 𝑥𝑥𝐴 𝑡𝐵
2 nfv 1912 . . . . . . . 8 𝑥 𝑡𝑧
3 eleq2 2828 . . . . . . . 8 (𝑧 = 𝐵 → (𝑡𝑧𝑡𝐵))
4 vex 3482 . . . . . . . . 9 𝑧 ∈ V
54a1i 11 . . . . . . . 8 (∀𝑥𝐴 𝑡𝐵𝑧 ∈ V)
6 rspa 3246 . . . . . . . 8 ((∀𝑥𝐴 𝑡𝐵𝑥𝐴) → 𝑡𝐵)
71, 2, 3, 5, 6elabreximd 32538 . . . . . . 7 ((∀𝑥𝐴 𝑡𝐵𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑡𝑧)
87ex 412 . . . . . 6 (∀𝑥𝐴 𝑡𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
98alrimiv 1925 . . . . 5 (∀𝑥𝐴 𝑡𝐵 → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
109adantl 481 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑥𝐴 𝑡𝐵) → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
11 nfra1 3282 . . . . . 6 𝑥𝑥𝐴 𝐵𝑉
122nfci 2891 . . . . . . . . 9 𝑥𝑧
13 nfre1 3283 . . . . . . . . . 10 𝑥𝑥𝐴 𝑦 = 𝐵
1413nfab 2909 . . . . . . . . 9 𝑥{𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
1512, 14nfel 2918 . . . . . . . 8 𝑥 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
1615, 2nfim 1894 . . . . . . 7 𝑥(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)
1716nfal 2322 . . . . . 6 𝑥𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)
1811, 17nfan 1897 . . . . 5 𝑥(∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
19 rspa 3246 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝑉𝑥𝐴) → 𝐵𝑉)
2019elexd 3502 . . . . . . . 8 ((∀𝑥𝐴 𝐵𝑉𝑥𝐴) → 𝐵 ∈ V)
2120adantlr 715 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝐵 ∈ V)
22 simplr 769 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
23 rspe 3247 . . . . . . . . . . . 12 ((𝑥𝐴𝑦 = 𝐵) → ∃𝑥𝐴 𝑦 = 𝐵)
24 tbtru 1545 . . . . . . . . . . . 12 (∃𝑥𝐴 𝑦 = 𝐵 ↔ (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))
2523, 24sylib 218 . . . . . . . . . . 11 ((𝑥𝐴𝑦 = 𝐵) → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))
2625ex 412 . . . . . . . . . 10 (𝑥𝐴 → (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
2726alrimiv 1925 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
2827adantl 481 . . . . . . . 8 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
29 elabgt 3672 . . . . . . . . 9 ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ⊤))
30 tbtru 1545 . . . . . . . . 9 (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ⊤))
3129, 30sylibr 234 . . . . . . . 8 ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3221, 28, 31syl2anc 584 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
33 eleq1 2827 . . . . . . . . . . 11 (𝑧 = 𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
3433, 3imbi12d 344 . . . . . . . . . 10 (𝑧 = 𝐵 → ((𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧) ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵)))
3534spcgv 3596 . . . . . . . . 9 (𝐵 ∈ V → (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵)))
3635imp 406 . . . . . . . 8 ((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵))
3736imp 406 . . . . . . 7 (((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑡𝐵)
3821, 22, 32, 37syl21anc 838 . . . . . 6 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝑡𝐵)
3938ex 412 . . . . 5 ((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) → (𝑥𝐴𝑡𝐵))
4018, 39ralrimi 3255 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) → ∀𝑥𝐴 𝑡𝐵)
4110, 40impbida 801 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑥𝐴 𝑡𝐵 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)))
4241abbidv 2806 . 2 (∀𝑥𝐴 𝐵𝑉 → {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)})
43 df-iin 4999 . . 3 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵}
4443a1i 11 . 2 (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵})
45 df-int 4952 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)}
4645a1i 11 . 2 (∀𝑥𝐴 𝐵𝑉 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)})
4742, 44, 463eqtr4d 2785 1 (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wtru 1538  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478   cint 4951   ciin 4997
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-int 4952  df-iin 4999
This theorem is referenced by:  intimafv  32726
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