Theorem iunpreima 32654

Description: Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)

Assertion

Ref	Expression
iunpreima	⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

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.

Step	Hyp	Ref	Expression
1		eliun 4926	. . . . 5 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵)
2	1	a1i 11	. . . 4 ⊢ (Fun 𝐹 → ((𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵))
3	2	rabbidv 3398	. . 3 ⊢ (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵})
4		funfn 6516	. . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
5		fncnvima2 7003	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵})
6	4, 5	sylbi 218	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵})
7		iunrab 4983	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵}
8	7	a1i 11	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵})
9	3, 6, 8	3eqtr4d 2784	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
10		fncnvima2 7003	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
11	4, 10	sylbi 218	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
12	11	iuneq2d 4953	. 2 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
13	9, 12	eqtr4d 2777	1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  {crab 3391  ∪ ciun 4922  ^◡ccnv 5618  dom cdm 5619   “ cima 5622  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  elrspunidl  33512