Theorem unipreima 32133

Description: Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)

Assertion

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

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem unipreima

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 6579	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		r19.42v 3189	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥))
3	2	bicomi 223	. . . . . 6 ⊢ ((𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥))
4	3	a1i 11	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥)))
5		eluni2 4913	. . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥)
6	5	anbi2i 622	. . . . . 6 ⊢ ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥))
7	6	a1i 11	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥)))
8		elpreima 7060	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥)))
9	8	rexbidv 3177	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥)))
10	4, 7, 9	3bitr4d 310	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥)))
11		elpreima 7060	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴)))
12		eliun 5002	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥))
13	12	a1i 11	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥)))
14	10, 11, 13	3bitr4d 310	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)))
15	14	eqrdv 2729	. 2 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))
16	1, 15	sylbi 216	1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  ∪ cuni 4909  ∪ 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 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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:  imambfm  33556  dstrvprob  33765