Theorem fmpt 5793

Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
fmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)

Assertion

Ref	Expression
fmpt	⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem fmpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmpt.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
2	1	fnmpt 5456	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹 Fn 𝐴)
3	1	rnmpt 4978	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶}
4		r19.29 2668	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶))
5		eleq1 2292	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
6	5	biimparc 299	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶) → 𝑦 ∈ 𝐵)
7	6	rexlimivw 2644	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶) → 𝑦 ∈ 𝐵)
8	4, 7	syl 14	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝑦 ∈ 𝐵)
9	8	ex 115	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → 𝑦 ∈ 𝐵))
10	9	abssdv 3299	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶} ⊆ 𝐵)
11	3, 10	eqsstrid 3271	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran 𝐹 ⊆ 𝐵)
12		df-f 5328	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
13	2, 11, 12	sylanbrc 417	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹:𝐴⟶𝐵)
14		fimacnv 5772	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴)
15	1	mptpreima 5228	. . . 4 ⊢ (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵}
16	14, 15	eqtr3di 2277	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵})
17		rabid2 2708	. . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵} ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
18	16, 17	sylib 122	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
19	13, 18	impbii 126	1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  {crab 2512   ⊆ wss 3198   ↦ cmpt 4148  ^◡ccnv 4722  ran crn 4724   “ cima 4726   Fn wfn 5319  ⟶wf 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332

This theorem is referenced by:  f1ompt  5794  fmpti  5795  fvmptelcdm  5796  fmptd  5797  fmptdf  5800  rnmptss  5804  f1oresrab  5808  idref  5892  f1mpt  5907  f1stres  6317  f2ndres  6318  fmpox  6360  fmpoco  6376  iunon  6445  mptelixpg  6898  dom2lem  6940  uzf  9748  ccatalpha  11180  pcmptcl  12905  gsumfzmhm2  13921  upxp  14986  txdis1cn  14992  cnmpt11  14997  cnmpt21  15005  fsumcncntop  15281  cncfmpt1f  15312  mulcncflem  15321  mulcncf  15322  cnmptlimc  15388  sincn  15483  coscn  15484  lgseisenlem3  15791