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Theorem iinabrex 31788
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 1918 . . . . . . . 8 𝑥 𝑡𝑧
3 eleq2 2823 . . . . . . . 8 (𝑧 = 𝐵 → (𝑡𝑧𝑡𝐵))
4 vex 3479 . . . . . . . . 9 𝑧 ∈ V
54a1i 11 . . . . . . . 8 (∀𝑥𝐴 𝑡𝐵𝑧 ∈ V)
6 rspa 3246 . . . . . . . 8 ((∀𝑥𝐴 𝑡𝐵𝑥𝐴) → 𝑡𝐵)
71, 2, 3, 5, 6elabreximd 31735 . . . . . . 7 ((∀𝑥𝐴 𝑡𝐵𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑡𝑧)
87ex 414 . . . . . 6 (∀𝑥𝐴 𝑡𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
98alrimiv 1931 . . . . 5 (∀𝑥𝐴 𝑡𝐵 → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
109adantl 483 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑥𝐴 𝑡𝐵) → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
11 nfra1 3282 . . . . . 6 𝑥𝑥𝐴 𝐵𝑉
122nfci 2887 . . . . . . . . 9 𝑥𝑧
13 nfre1 3283 . . . . . . . . . 10 𝑥𝑥𝐴 𝑦 = 𝐵
1413nfab 2910 . . . . . . . . 9 𝑥{𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
1512, 14nfel 2918 . . . . . . . 8 𝑥 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
1615, 2nfim 1900 . . . . . . 7 𝑥(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)
1716nfal 2317 . . . . . 6 𝑥𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)
1811, 17nfan 1903 . . . . 5 𝑥(∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
19 rspa 3246 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝑉𝑥𝐴) → 𝐵𝑉)
2019elexd 3495 . . . . . . . 8 ((∀𝑥𝐴 𝐵𝑉𝑥𝐴) → 𝐵 ∈ V)
2120adantlr 714 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝐵 ∈ V)
22 simplr 768 . . . . . . 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 1550 . . . . . . . . . . . 12 (∃𝑥𝐴 𝑦 = 𝐵 ↔ (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))
2523, 24sylib 217 . . . . . . . . . . 11 ((𝑥𝐴𝑦 = 𝐵) → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))
2625ex 414 . . . . . . . . . 10 (𝑥𝐴 → (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
2726alrimiv 1931 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
2827adantl 483 . . . . . . . 8 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
29 elabgt 3662 . . . . . . . . 9 ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ⊤))
30 tbtru 1550 . . . . . . . . 9 (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ⊤))
3129, 30sylibr 233 . . . . . . . 8 ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3221, 28, 31syl2anc 585 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
33 eleq1 2822 . . . . . . . . . . 11 (𝑧 = 𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
3433, 3imbi12d 345 . . . . . . . . . 10 (𝑧 = 𝐵 → ((𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧) ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵)))
3534spcgv 3587 . . . . . . . . 9 (𝐵 ∈ V → (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵)))
3635imp 408 . . . . . . . 8 ((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵))
3736imp 408 . . . . . . 7 (((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑡𝐵)
3821, 22, 32, 37syl21anc 837 . . . . . 6 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝑡𝐵)
3938ex 414 . . . . 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 800 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑥𝐴 𝑡𝐵 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)))
4241abbidv 2802 . 2 (∀𝑥𝐴 𝐵𝑉 → {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)})
43 df-iin 5000 . . 3 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵}
4443a1i 11 . 2 (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵})
45 df-int 4951 . . 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 2783 1 (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wtru 1543  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475   cint 4950   ciin 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-v 3477  df-int 4951  df-iin 5000
This theorem is referenced by:  intimafv  31920
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