Theorem fvelimad 6901

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 6023	. . . . 5 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶))
3	2	ibi 268	. . . 4 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶)
5		nfv 1921	. . . 4 ⊢ Ⅎ𝑦𝜑
6		nfre1 3265	. . . 4 ⊢ Ⅎ𝑦∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶
7		vex 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ V)
9	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝐶 ∈ (𝐹 “ 𝐵))
10		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦𝐹𝐶)
11	8, 9, 10	breldmd 5861	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
12		fvelimad.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13	12	fndmd 6597	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	13	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
15	11, 14	eleqtrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐴)
16	15	3adant2 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐴)
17		simp2 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐵)
18	16, 17	elind 4136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ (𝐴 ∩ 𝐵))
19		fnfun 6592	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
20	12, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
21	20	3ad2ant1 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → Fun 𝐹)
22		simp3 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦𝐹𝐶)
23		funbrfv 6882	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑦𝐹𝐶 → (𝐹‘𝑦) = 𝐶))
24	21, 22, 23	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → (𝐹‘𝑦) = 𝐶)
25		rspe 3230	. . . . . 6 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝐵) ∧ (𝐹‘𝑦) = 𝐶) → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
26	18, 24, 25	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
27	26	3exp 1125	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)))
28	5, 6, 27	rexlimd 3247	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶))
29	4, 28	mpd 15	. 2 ⊢ (𝜑 → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
30		nfv 1921	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐶
31		fvelimad.x	. . . . 5 ⊢ Ⅎ𝑥𝐹
32		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑦
33	31, 32	nffv 6844	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
34	33	nfeq1 2917	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐶
35		fveqeq2 6843	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = 𝐶 ↔ (𝐹‘𝑦) = 𝐶))
36	30, 34, 35	cbvrexw 3283	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶 ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
37	29, 36	sylibr 235	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Ⅎwnfc 2887  ∃wrex 3064  Vcvv 3432   ∩ cin 3889   class class class wbr 5079  dom cdm 5625   “ cima 5628  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  cyc3evpm  33238  cycpmgcl  33241  cycpmconjslem2  33243  cyc3conja  33245  limsupmnflem  46170  liminfvalxr  46233