Theorem unpreima 5804

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 5384	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elpreima 5799	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵))))
3		elun 3362	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵) ↔ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵))
4	3	anbi2i 457	. . . . . 6 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵)))
5		andi 826	. . . . . 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 184	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))
7		elun 3362	. . . . . 6 ⊢ (𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)) ↔ (𝑥 ∈ (◡𝐹 “ 𝐴) ∨ 𝑥 ∈ (◡𝐹 “ 𝐵)))
8		elpreima 5799	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴)))
9		elpreima 5799	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))
10	8, 9	orbi12d 801	. . . . . 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 192	. . . . 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 199	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))))
13	2, 12	bitrd 188	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ (𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))))
14	13	eqrdv 2232	. 2 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))
15	1, 14	sylbi 121	1 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2205   ∪ cun 3211  ^◡ccnv 4750  dom cdm 4751   “ cima 4754  Fun wfun 5348   Fn wfn 5349  ‘cfv 5354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by: (None)