Theorem fvelimad 6856

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 5974	. . . . 5 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶))
3	2	ibi 266	. . . 4 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝑦𝐹𝐶)
5		nfv 1913	. . . 4 ⊢ Ⅎ𝑦𝜑
6		nfre1 3267	. . . 4 ⊢ Ⅎ𝑦∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶
7		vex 3438	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ V)
9	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝐶 ∈ (𝐹 “ 𝐵))
10		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦𝐹𝐶)
11	8, 9, 10	breldmd 5825	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
12		fvelimad.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13	12	fndmd 6557	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
15	11, 14	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐴)
16	15	3adant2 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐴)
17		simp2 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ 𝐵)
18	16, 17	elind 4131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦 ∈ (𝐴 ∩ 𝐵))
19		fnfun 6552	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
20	12, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
21	20	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → Fun 𝐹)
22		simp3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → 𝑦𝐹𝐶)
23		funbrfv 6840	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑦𝐹𝐶 → (𝐹‘𝑦) = 𝐶))
24	21, 22, 23	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → (𝐹‘𝑦) = 𝐶)
25		rspe 3265	. . . . . 6 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝐵) ∧ (𝐹‘𝑦) = 𝐶) → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
26	18, 24, 25	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
27	26	3exp 1117	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)))
28	5, 6, 27	rexlimd 3278	. . 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 1913	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = 𝐶
31		fvelimad.x	. . . . 5 ⊢ Ⅎ𝑥𝐹
32		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑦
33	31, 32	nffv 6802	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
34	33	nfeq1 2917	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = 𝐶
35		fveqeq2 6801	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = 𝐶 ↔ (𝐹‘𝑦) = 𝐶))
36	30, 34, 35	cbvrexw 3376	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶 ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑦) = 𝐶)
37	29, 36	sylibr 233	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1537   ∈ wcel 2101  Ⅎwnfc 2882  ∃wrex 3068  Vcvv 3434   ∩ cin 3888   class class class wbr 5077  dom cdm 5591   “ cima 5594  Fun wfun 6441   Fn wfn 6442  ‘cfv 6447

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 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-fv 6455

This theorem is referenced by:  cyc3evpm  31445  cycpmgcl  31448  cycpmconjslem2  31450  cyc3conja  31452  limsupmnflem  43296  liminfvalxr  43359