Theorem unpreima 7002

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 6516	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elpreima 6997	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵))))
3		elun 4102	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵) ↔ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵))
4	3	anbi2i 623	. . . . . 6 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵)))
5		andi 1009	. . . . . 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 4102	. . . . . 6 ⊢ (𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)) ↔ (𝑥 ∈ (◡𝐹 “ 𝐴) ∨ 𝑥 ∈ (◡𝐹 “ 𝐵)))
8		elpreima 6997	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴)))
9		elpreima 6997	. . . . . . 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 217	1 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ∪ cun 3896  ^◡ccnv 5618  dom cdm 5619   “ cima 5622  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  sibfof  34374