Theorem fvelima2 45169

Description: Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Assertion

Ref	Expression
fvelima2	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Proof of Theorem fvelima2

Step	Hyp	Ref	Expression
1		elimag 6093	. . . 4 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → (𝐵 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵))
2	1	ibi 267	. . 3 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵)
3		df-rex 3077	. . 3 ⊢ (∃𝑥 ∈ 𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵))
4	2, 3	sylib 218	. 2 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵))
5		fnbr 6687	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝐵) → 𝑥 ∈ 𝐴)
6	5	adantrl 715	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ 𝐴)
7		simprl 770	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ 𝐶)
8	6, 7	elind 4223	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴 ∩ 𝐶))
9		fnfun 6679	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
10		funbrfv 6971	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹‘𝑥) = 𝐵))
11	10	imp 406	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥𝐹𝐵) → (𝐹‘𝑥) = 𝐵)
12	9, 11	sylan 579	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝐵) → (𝐹‘𝑥) = 𝐵)
13	12	adantrl 715	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → (𝐹‘𝑥) = 𝐵)
14	8, 13	jca 511	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))
15	14	ex 412	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵)))
16	15	eximdv 1916	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵)))
17	16	imp 406	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))
18		df-rex 3077	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))
19	17, 18	sylibr 234	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵)
20	4, 19	sylan2 592	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076   ∩ cin 3975   class class class wbr 5166   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  limsupresxr  45687  liminfresxr  45688  liminfvalxr  45704