Theorem unipreima 32845

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 6551	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		r19.42v 3194	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥))
3	2	bicomi 226	. . . . . 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 4869	. . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥)
6	5	anbi2i 632	. . . . . 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 7039	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥)))
9	8	rexbidv 3186	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥)))
10	4, 7, 9	3bitr4d 313	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥)))
11		elpreima 7039	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴)))
12		eliun 4953	. . . . 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 313	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)))
15	14	eqrdv 2760	. 2 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))
16	1, 15	sylbi 219	1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086  ∪ cuni 4865  ∪ ciun 4949  ^◡ccnv 5646  dom cdm 5647   “ cima 5650  Fun wfun 6515   Fn wfn 6516  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529

This theorem is referenced by:  imambfm  34559  dstrvprob  34769