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Theorem fmpt 6876

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 6490	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹 Fn 𝐴)
3	1	rnmpt 5829	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶}
4		r19.29 3256	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶))
5		eleq1 2902	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
6	5	biimparc 482	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦 = 𝐶) → 𝑦 ∈ 𝐵)
7	6	rexlimivw 3284	. . . . . . 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 415	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → 𝑦 ∈ 𝐵))
10	9	abssdv 4047	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶} ⊆ 𝐵)
11	3, 10	eqsstrid 4017	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran 𝐹 ⊆ 𝐵)
12		df-f 6361	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
13	2, 11, 12	sylanbrc 585	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹:𝐴⟶𝐵)
14	1	mptpreima 6094	. . . 4 ⊢ (^◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵}
15		fimacnv 6841	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)
16	14, 15	syl5reqr 2873	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵})
17		rabid2 3383	. . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ 𝐵} ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
18	16, 17	sylib 220	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
19	13, 18	impbii 211	1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ↦ cmpt 5148 ^◡ccnv 5556 ran crn 5558 “ cima 5560 Fn wfn 6352 ⟶wf 6353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365

This theorem is referenced by: f1ompt 6877 fmpti 6878 fvmptelrn 6879 fmptd 6880 fmptdf 6883 rnmptss 6888 f1oresrab 6891 idref 6910 f1mpt 7021 f1stres 7715 f2ndres 7716 fmpox 7767 fmpoco 7792 onoviun 7982 onnseq 7983 mptelixpg 8501 dom2lem 8551 iinfi 8883 cantnfrescl 9141 acni2 9474 acnlem 9476 dfac4 9550 dfacacn 9569 fin23lem28 9764 axdc2lem 9872 axcclem 9881 ac6num 9903 uzf 12249 ccatalpha 13949 repsf 14137 rlim2 14855 rlimi 14872 o1fsum 15170 ackbijnn 15185 pcmptcl 16229 vdwlem11 16329 ismon2 17006 isepi2 17013 yonedalem3b 17531 smndex1gbas 18069 efgsf 18857 gsummhm2 19061 gsummptcl 19089 gsummptfif1o 19090 gsummptfzcl 19091 gsumcom2 19097 gsummptnn0fz 19108 issrngd 19634 subrgasclcl 20281 evl1sca 20499 ipcl 20779 mavmulcl 21158 m2detleiblem3 21240 m2detleiblem4 21241 iinopn 21512 ordtrest2 21814 iscnp2 21849 discmp 22008 2ndcdisj 22066 ptunimpt 22205 pttopon 22206 ptcnplem 22231 upxp 22233 txdis1cn 22245 cnmpt11 22273 cnmpt21 22281 cnmptkp 22290 cnmptk1 22291 cnmpt1k 22292 cnmptkk 22293 cnmptk1p 22295 qtopeu 22326 uzrest 22507 txflf 22616 clsnsg 22720 tgpconncomp 22723 tsmsf1o 22755 prdsmet 22982 fsumcn 23480 cncfmpt1f 23523 iccpnfcnv 23550 lebnumlem1 23567 copco 23624 pcoass 23630 bcth3 23936 voliun 24157 i1f1lem 24292 iblcnlem 24391 limcvallem 24471 ellimc2 24477 cnmptlimc 24490 dvle 24606 dvfsumle 24620 dvfsumge 24621 dvfsumabs 24622 dvfsumlem2 24626 itgsubstlem 24647 sincn 25034 coscn 25035 rlimcxp 25553 harmonicbnd 25583 harmonicbnd2 25584 lgamgulmlem6 25613 sqff1o 25761 lgseisenlem3 25955 fmptdF 30403 ordtrest2NEW 31168 ddemeas 31497 eulerpartgbij 31632 0rrv 31711 reprpmtf1o 31899 subfacf 32424 tailf 33725 fdc 35022 heiborlem5 35095 elrfirn2 39300 mptfcl 39324 mzpexpmpt 39349 mzpsubst 39352 rabdiophlem1 39405 rabdiophlem2 39406 pw2f1ocnv 39641 refsumcn 41294 fompt 41460 fmptf 41516 fprodcnlem 41887 dvsinax 42204 itgsubsticclem 42267 fargshiftf 43607 isomuspgrlem2b 44001

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