Theorem fvelima2 6894

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 6031	. . . 4 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → (𝐵 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵))
2	1	ibi 267	. . 3 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵)
3		df-rex 3063	. . 3 ⊢ (∃𝑥 ∈ 𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵))
4	2, 3	sylib 218	. 2 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵))
5		fnbr 6608	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝐵) → 𝑥 ∈ 𝐴)
6	5	adantrl 717	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ 𝐴)
7		simprl 771	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ 𝐶)
8	6, 7	elind 4154	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴 ∩ 𝐶))
9		fnfun 6600	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
10		funbrfv 6890	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹‘𝑥) = 𝐵))
11	10	imp 406	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥𝐹𝐵) → (𝐹‘𝑥) = 𝐵)
12	9, 11	sylan 581	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝐵) → (𝐹‘𝑥) = 𝐵)
13	12	adantrl 717	. . . . . . 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 1919	. . . 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 3063	. . 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 594	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3902   class class class wbr 5100   “ cima 5635  Fun wfun 6494   Fn wfn 6495  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  exsslsb  33774  limsupresxr  46124  liminfresxr  46125  liminfvalxr  46141