Theorem iunpreima 31796

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 5002	. . . . 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 3441	. . 3 ⊢ (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵})
4		funfn 6579	. . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
5		fncnvima2 7063	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵})
6	4, 5	sylbi 216	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵})
7		iunrab 5056	. . . 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 2783	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
10		fncnvima2 7063	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
11	4, 10	sylbi 216	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
12	11	iuneq2d 5027	. 2 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
13	9, 12	eqtr4d 2776	1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  {crab 3433  ∪ ciun 4998  ^◡ccnv 5676  dom cdm 5677   “ cima 5680  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544

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  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552

This theorem is referenced by:  elrspunidl  32546