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Theorem fmpt 7144

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 6720	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹 Fn 𝐴)
3	1	rnmpt 5980	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶}
4		r19.29 3120	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶))
5		eleq1 2832	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
6	5	biimparc 479	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶) → 𝑦 ∈ 𝐵)
7	6	rexlimivw 3157	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶) → 𝑦 ∈ 𝐵)
8	4, 7	syl 17	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝑦 ∈ 𝐵)
9	8	ex 412	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → 𝑦 ∈ 𝐵))
10	9	abssdv 4091	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶} ⊆ 𝐵)
11	3, 10	eqsstrid 4057	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran 𝐹 ⊆ 𝐵)
12		df-f 6577	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
13	2, 11, 12	sylanbrc 582	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹:𝐴⟶𝐵)
14		fimacnv 6769	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)
15	1	mptpreima 6269	. . . 4 ⊢ (^◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵}
16	14, 15	eqtr3di 2795	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵})
17		rabid2 3478	. . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵} ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
18	16, 17	sylib 218	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
19	13, 18	impbii 209	1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {cab 2717 ∀wral 3067 ∃wrex 3076 {crab 3443 ⊆ wss 3976 ↦ cmpt 5249 ^◡ccnv 5699 ran crn 5701 “ cima 5703 Fn wfn 6568 ⟶wf 6569

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-fun 6575 df-fn 6576 df-f 6577

This theorem is referenced by: f1ompt 7145 fmpti 7146 fvmptelcdm 7147 fmptd 7148 fmptdf 7151 fompt 7152 rnmptss 7157 f1oresrab 7161 idref 7180 f1mpt 7298 f1stres 8054 f2ndres 8055 fmpox 8108 fmpoco 8136 onoviun 8399 onnseq 8400 mptelixpg 8993 dom2lem 9052 iinfi 9486 cantnfrescl 9745 acni2 10115 acnlem 10117 dfac4 10191 dfacacn 10211 fin23lem28 10409 axdc2lem 10517 axcclem 10526 ac6num 10548 uzf 12906 ccatalpha 14641 repsf 14821 rlim2 15542 rlimi 15559 o1fsum 15861 ackbijnn 15876 pcmptcl 16938 vdwlem11 17038 ismon2 17795 isepi2 17802 yonedalem3b 18349 smndex1gbas 18937 efgsf 19771 gsummhm2 19981 gsummptcl 20009 gsummptfif1o 20010 gsummptfzcl 20011 gsumcom2 20017 gsummptnn0fz 20028 issrngd 20878 ipcl 21674 subrgasclcl 22114 evl1sca 22359 mavmulcl 22574 m2detleiblem3 22656 m2detleiblem4 22657 iinopn 22929 ordtrest2 23233 iscnp2 23268 discmp 23427 2ndcdisj 23485 ptunimpt 23624 pttopon 23625 ptcnplem 23650 upxp 23652 txdis1cn 23664 cnmpt11 23692 cnmpt21 23700 cnmptkp 23709 cnmptk1 23710 cnmpt1k 23711 cnmptkk 23712 cnmptk1p 23714 qtopeu 23745 uzrest 23926 txflf 24035 clsnsg 24139 tgpconncomp 24142 tsmsf1o 24174 prdsmet 24401 fsumcn 24913 cncfmpt1f 24959 iccpnfcnv 24994 lebnumlem1 25012 copco 25070 pcoass 25076 bcth3 25384 voliun 25608 i1f1lem 25743 iblcnlem 25844 limcvallem 25926 ellimc2 25932 cnmptlimc 25945 dvle 26066 dvfsumle 26080 dvfsumleOLD 26081 dvfsumge 26082 dvfsumabs 26083 dvfsumlem2 26087 dvfsumlem2OLD 26088 itgsubstlem 26109 sincn 26506 coscn 26507 rlimcxp 27035 harmonicbnd 27065 harmonicbnd2 27066 lgamgulmlem6 27095 sqff1o 27243 lgseisenlem3 27439 fmptdF 32674 ordtrest2NEW 33869 ddemeas 34200 eulerpartgbij 34337 0rrv 34416 reprpmtf1o 34603 subfacf 35143 tailf 36341 fdc 37705 heiborlem5 37775 3factsumint 41982 dvle2 42029 fmpocos 42229 elrfirn2 42652 mptfcl 42676 mzpexpmpt 42701 mzpsubst 42704 rabdiophlem1 42757 rabdiophlem2 42758 pw2f1ocnv 42994 refsumcn 44930 fmptf 45147 fmptff 45179 fprodcnlem 45520 dvsinax 45834 itgsubsticclem 45896 fargshiftf 47314

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