Theorem fvelima2 43951

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 6062	. . . 4 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → (𝐵 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵))
2	1	ibi 267	. . 3 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵)
3		df-rex 3072	. . 3 ⊢ (∃𝑥 ∈ 𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵))
4	2, 3	sylib 217	. 2 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵))
5		fnbr 6655	. . . . . . . . 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 4194	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴 ∩ 𝐶))
9		fnfun 6647	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
10		funbrfv 6940	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹‘𝑥) = 𝐵))
11	10	imp 408	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥𝐹𝐵) → (𝐹‘𝑥) = 𝐵)
12	9, 11	sylan 581	. . . . . . . 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 513	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))
15	14	ex 414	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵)))
16	15	eximdv 1921	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵)))
17	16	imp 408	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))
18		df-rex 3072	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))
19	17, 18	sylibr 233	. 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 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∃wrex 3071   ∩ cin 3947   class class class wbr 5148   “ 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-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:  limsupresxr  44469  liminfresxr  44470  liminfvalxr  44486