Theorem funimass4 6816

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
funimass4	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

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

Proof of Theorem funimass4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3903	. . 3 ⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵))
2		vex 3426	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	2	elima 5963	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)
4		eqcom 2745	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
5		ssel 3910	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹))
6		funbrfvb 6806	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
7	6	ex 412	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)))
8	5, 7	syl9 77	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ dom 𝐹 → (Fun 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))))
9	8	imp31 417	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
10	4, 9	syl5bb 282	. . . . . . . . 9 ⊢ (((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
11	10	rexbidva 3224	. . . . . . . 8 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
12	3, 11	bitr4id 289	. . . . . . 7 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
13	12	imbi1d 341	. . . . . 6 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)))
14		r19.23v 3207	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))
15	13, 14	bitr4di 288	. . . . 5 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)))
16	15	albidv 1924	. . . 4 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)))
17		ralcom4 3161	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))
18		fvex 6769	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
19		eleq1 2826	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵))
20	18, 19	ceqsalv 3457	. . . . . 6 ⊢ (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (𝐹‘𝑥) ∈ 𝐵)
21	20	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
22	17, 21	bitr3i 276	. . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
23	16, 22	bitrdi 286	. . 3 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
24	1, 23	syl5bb 282	. 2 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
25	24	ancoms 458	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883   class class class wbr 5070  dom cdm 5580   “ cima 5583  Fun wfun 6412  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  funimass3  6913  funimass5  6914  funconstss  6915  funimassov  7427  fnwelem  7943  cnfcomlem  9387  dfac12lem2  9831  ackbij1b  9926  wunom  10407  phimullem  16408  frmdss2  18417  cntzmhm2  18861  dprd2da  19560  frlmsslsp  20913  1stckgenlem  22612  txcnp  22679  ptcnplem  22680  xkopt  22714  xkoinjcn  22746  tgqtop  22771  uzrest  22956  cnflf2  23062  lmflf  23064  txflf  23065  cnextcn  23126  ghmcnp  23174  ucnima  23341  metcnp  23603  tcphcph  24306  ovolficcss  24538  opnmbllem  24670  ellimc2  24946  ellimc3  24948  deg1n0ima  25159  dvloglem  25708  logf1o2  25710  dchrghm  26309  upgrreslem  27574  umgrreslem  27575  xrofsup  30992  eulerpartlemd  32233  erdszelem2  33054  cvmlift3lem7  33187  mclsax  33431  madebdayim  33997  filnetlem4  34497  poimir  35737  opnmbllem0  35740  cnres2  35848  icccncfext  43318