Theorem fvelimab 5477

Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)

Assertion

Ref	Expression
fvelimab	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

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

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fvelimab

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2697	. . . 4 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → 𝐶 ∈ V)
2	1	anim2i 339	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ (𝐹 “ 𝐵)) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
3		ssel2 3092	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐴)
4		funfvex 5438	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢 ∈ dom 𝐹) → (𝐹‘𝑢) ∈ V)
5	4	funfni 5223	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ V)
6	3, 5	sylan2 284	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑢 ∈ 𝐵)) → (𝐹‘𝑢) ∈ V)
7	6	anassrs 397	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝐹‘𝑢) ∈ V)
8		eleq1 2202	. . . . . 6 ⊢ ((𝐹‘𝑢) = 𝐶 → ((𝐹‘𝑢) ∈ V ↔ 𝐶 ∈ V))
9	7, 8	syl5ibcom 154	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐵) → ((𝐹‘𝑢) = 𝐶 → 𝐶 ∈ V))
10	9	rexlimdva 2549	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶 → 𝐶 ∈ V))
11	10	imdistani 441	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
12		eleq1 2202	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵)))
13		eqeq2 2149	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((𝐹‘𝑢) = 𝑣 ↔ (𝐹‘𝑢) = 𝐶))
14	13	rexbidv 2438	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣 ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
15	12, 14	bibi12d 234	. . . . . 6 ⊢ (𝑣 = 𝐶 → ((𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣) ↔ (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶)))
16	15	imbi2d 229	. . . . 5 ⊢ (𝑣 = 𝐶 → (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣)) ↔ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))))
17		fnfun 5220	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
18	17	adantr 274	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → Fun 𝐹)
19		fndm 5222	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
20	19	sseq2d 3127	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
21	20	biimpar 295	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
22		dfimafn 5470	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
23	18, 21, 22	syl2anc 408	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
24	23	abeq2d 2252	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣))
25	16, 24	vtoclg 2746	. . . 4 ⊢ (𝐶 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶)))
26	25	impcom 124	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
27	2, 11, 26	pm5.21nd 901	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
28		fveq2 5421	. . . 4 ⊢ (𝑢 = 𝑥 → (𝐹‘𝑢) = (𝐹‘𝑥))
29	28	eqeq1d 2148	. . 3 ⊢ (𝑢 = 𝑥 → ((𝐹‘𝑢) = 𝐶 ↔ (𝐹‘𝑥) = 𝐶))
30	29	cbvrexv 2655	. 2 ⊢ (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶 ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)
31	27, 30	syl6bb 195	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125  ∃wrex 2417  Vcvv 2686   ⊆ wss 3071  dom cdm 4539   “ cima 4542  Fun wfun 5117   Fn wfn 5118  ‘cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131

This theorem is referenced by:  ssimaex  5482  foima2  5653  rexima  5656  ralima  5657  f1elima  5674  ovelimab  5921