Theorem fvelimad 6978

Description: Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
fvelimad.x	⊢ Ⅎ𝑥𝐹
fvelimad.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
fvelimad.c	⊢ (𝜑 → 𝐶 ∈ (𝐹 “ 𝐵))

Assertion

Ref	Expression
fvelimad	⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶)

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

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem fvelimad

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvelimad.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐹 “ 𝐵))
2		elimag 6073	. . . . 5 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶))
3	2	ibi 270	. . . 4 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶)
4	1, 3	syl 18	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶)
5		nfv 1949	. . . 4 ⊢ Ⅎ𝑦𝜑
6		nfre1 3291	. . . 4 ⊢ Ⅎ𝑦∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶
7		vex 3458	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ V)
9	1	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝐶 ∈ (𝐹 “ 𝐵))
10		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦𝐹𝐶)
11	8, 9, 10	breldmd 5902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
12		fvelimad.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13	12	fndmd 6669	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
15	11, 14	eleqtrd 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐴)
16	15	3adant2 1149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐴)
17		simp2 1155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐵)
18	16, 17	elind 4100	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ (𝐴 ∩ 𝐵))
19		fnfun 6664	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
20	12, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
21	20	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → Fun 𝐹)
22		simp3 1156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦𝐹𝐶)
23		funbrfv 6959	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑦𝐹𝐶 → (𝐹‘𝑦) = 𝐶))
24	21, 22, 23	sylc 66	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → (𝐹‘𝑦) = 𝐶)
25		rspe 3256	. . . . . 6 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝐵) ∧ (𝐹‘𝑦) = 𝐶) → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
26	18, 24, 25	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
27	26	3exp 1137	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)))
28	5, 6, 27	rexlimd 3273	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶))
29	4, 28	mpd 16	. 2 ⊢ (𝜑 → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
30		nfv 1949	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐶
31		fvelimad.x	. . . . 5 ⊢ Ⅎ𝑥𝐹
32		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑦
33	31, 32	nffv 6921	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
34	33	nfeq1 2941	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐶
35		fveqeq2 6920	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = 𝐶 ↔ (𝐹‘𝑦) = 𝐶))
36	30, 34, 35	cbvrexw 3309	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶 ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
37	29, 36	sylibr 237	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  ∃wrex 3090  Vcvv 3454   ∩ cin 4092   class class class wbr 5106  dom cdm 5656   “ cima 5659  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is used by:  cyc3evpm  33787  cycpmgcl  33790  cycpmconjslem2  33792  cyc3conja  33794  edompidlem  33940  limsupmnflem  46945  liminfvalxr  47008