Theorem unpreima 7062

Description: Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
unpreima	⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))

Proof of Theorem unpreima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 6576	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elpreima 7057	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵))))
3		elun 4148	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵) ↔ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵))
4	3	anbi2i 624	. . . . . 6 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵)))
5		andi 1007	. . . . . 6 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))
6	4, 5	bitri 275	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))
7		elun 4148	. . . . . 6 ⊢ (𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)) ↔ (𝑥 ∈ (◡𝐹 “ 𝐴) ∨ 𝑥 ∈ (◡𝐹 “ 𝐵)))
8		elpreima 7057	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴)))
9		elpreima 7057	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))
10	8, 9	orbi12d 918	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (◡𝐹 “ 𝐴) ∨ 𝑥 ∈ (◡𝐹 “ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))))
11	7, 10	bitrid 283	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))))
12	6, 11	bitr4id 290	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))))
13	2, 12	bitrd 279	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ (𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))))
14	13	eqrdv 2731	. 2 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))
15	1, 14	sylbi 216	1 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   ∪ cun 3946  ^◡ccnv 5675  dom cdm 5676   “ cima 5679  Fun wfun 6535   Fn wfn 6536  ‘cfv 6541

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-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

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-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-fv 6549

This theorem is referenced by:  sibfof  33328