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Theorem iunpreima 30332
Description: Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
Assertion
Ref Expression
iunpreima (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunpreima
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4888 . . . . 5 ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵)
21a1i 11 . . . 4 (Fun 𝐹 → ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵))
32rabbidv 3430 . . 3 (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵})
4 funfn 6358 . . . 4 (Fun 𝐹𝐹 Fn dom 𝐹)
5 fncnvima2 6812 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 𝑥𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵})
64, 5sylbi 220 . . 3 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵})
7 iunrab 4942 . . . 4 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵}
87a1i 11 . . 3 (Fun 𝐹 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵})
93, 6, 83eqtr4d 2846 . 2 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
10 fncnvima2 6812 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
114, 10sylbi 220 . . 3 (Fun 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
1211iuneq2d 4913 . 2 (Fun 𝐹 𝑥𝐴 (𝐹𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
139, 12eqtr4d 2839 1 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  wrex 3110  {crab 3113   ciun 4884  ccnv 5522  dom cdm 5523  cima 5526  Fun wfun 6322   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  elrspunidl  31018
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