Theorem unipreima 30391

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 6385	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		r19.42v 3350	. . . . . . 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 4842	. . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥)
6	5	anbi2i 624	. . . . . 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 6828	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥)))
9	8	rexbidv 3297	. . . . 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 6828	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴)))
12		eliun 4923	. . . . 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 2819	. 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 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  ∪ cuni 4838  ∪ ciun 4919  ^◡ccnv 5554  dom cdm 5555   “ cima 5558  Fun wfun 6349   Fn wfn 6350  ‘cfv 6355

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363

This theorem is referenced by:  imambfm  31520  dstrvprob  31729