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Theorem fmptco 6646
 Description: Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation (𝑥 + 2) and 𝐺 the equation (3∗𝑧) then (𝐺 ∘ 𝐹) has the equation (3∗(𝑥 + 2)). (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
fmptco.1 ((𝜑𝑥𝐴) → 𝑅𝐵)
fmptco.2 (𝜑𝐹 = (𝑥𝐴𝑅))
fmptco.3 (𝜑𝐺 = (𝑦𝐵𝑆))
fmptco.4 (𝑦 = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
fmptco (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑦,𝑅   𝜑,𝑥   𝑥,𝑆   𝑦,𝑇
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑇(𝑥)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fmptco
Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5874 . 2 Rel (𝐺𝐹)
2 mptrel 5481 . 2 Rel (𝑥𝐴𝑇)
3 fmptco.2 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑥𝐴𝑅))
4 fmptco.1 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑅𝐵)
53, 4fmpt3d 6635 . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
65ffund 6282 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
7 funbrfv 6480 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹𝑧) = 𝑢))
87imp 397 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
96, 8sylan 575 . . . . . . . . 9 ((𝜑𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
109eqcomd 2831 . . . . . . . 8 ((𝜑𝑧𝐹𝑢) → 𝑢 = (𝐹𝑧))
1110a1d 25 . . . . . . 7 ((𝜑𝑧𝐹𝑢) → (𝑢𝐺𝑤𝑢 = (𝐹𝑧)))
1211expimpd 447 . . . . . 6 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) → 𝑢 = (𝐹𝑧)))
1312pm4.71rd 558 . . . . 5 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
1413exbidv 2020 . . . 4 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
15 fvex 6446 . . . . . 6 (𝐹𝑧) ∈ V
16 breq2 4877 . . . . . . 7 (𝑢 = (𝐹𝑧) → (𝑧𝐹𝑢𝑧𝐹(𝐹𝑧)))
17 breq1 4876 . . . . . . 7 (𝑢 = (𝐹𝑧) → (𝑢𝐺𝑤 ↔ (𝐹𝑧)𝐺𝑤))
1816, 17anbi12d 624 . . . . . 6 (𝑢 = (𝐹𝑧) → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤)))
1915, 18ceqsexv 3459 . . . . 5 (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤))
20 funfvbrb 6579 . . . . . . . . 9 (Fun 𝐹 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
216, 20syl 17 . . . . . . . 8 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
225fdmd 6287 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐴)
2322eleq2d 2892 . . . . . . . 8 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
2421, 23bitr3d 273 . . . . . . 7 (𝜑 → (𝑧𝐹(𝐹𝑧) ↔ 𝑧𝐴))
253fveq1d 6435 . . . . . . . 8 (𝜑 → (𝐹𝑧) = ((𝑥𝐴𝑅)‘𝑧))
26 fmptco.3 . . . . . . . 8 (𝜑𝐺 = (𝑦𝐵𝑆))
27 eqidd 2826 . . . . . . . 8 (𝜑𝑤 = 𝑤)
2825, 26, 27breq123d 4887 . . . . . . 7 (𝜑 → ((𝐹𝑧)𝐺𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
2924, 28anbi12d 624 . . . . . 6 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤)))
30 nfcv 2969 . . . . . . . . 9 𝑥𝑧
31 nfv 2013 . . . . . . . . . 10 𝑥𝜑
32 nffvmpt1 6444 . . . . . . . . . . . 12 𝑥((𝑥𝐴𝑅)‘𝑧)
33 nfcv 2969 . . . . . . . . . . . 12 𝑥(𝑦𝐵𝑆)
34 nfcv 2969 . . . . . . . . . . . 12 𝑥𝑤
3532, 33, 34nfbr 4920 . . . . . . . . . . 11 𝑥((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤
36 nfcsb1v 3773 . . . . . . . . . . . 12 𝑥𝑧 / 𝑥𝑇
3736nfeq2 2985 . . . . . . . . . . 11 𝑥 𝑤 = 𝑧 / 𝑥𝑇
3835, 37nfbi 2006 . . . . . . . . . 10 𝑥(((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)
3931, 38nfim 1999 . . . . . . . . 9 𝑥(𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
40 fveq2 6433 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐴𝑅)‘𝑥) = ((𝑥𝐴𝑅)‘𝑧))
4140breq1d 4883 . . . . . . . . . . 11 (𝑥 = 𝑧 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
42 csbeq1a 3766 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑇 = 𝑧 / 𝑥𝑇)
4342eqeq2d 2835 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑤 = 𝑇𝑤 = 𝑧 / 𝑥𝑇))
4441, 43bibi12d 337 . . . . . . . . . 10 (𝑥 = 𝑧 → ((((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇) ↔ (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
4544imbi2d 332 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)) ↔ (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))))
46 vex 3417 . . . . . . . . . . . 12 𝑤 ∈ V
47 simpl 476 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑦 = 𝑅)
4847eleq1d 2891 . . . . . . . . . . . . . 14 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑦𝐵𝑅𝐵))
49 id 22 . . . . . . . . . . . . . . 15 (𝑢 = 𝑤𝑢 = 𝑤)
50 fmptco.4 . . . . . . . . . . . . . . 15 (𝑦 = 𝑅𝑆 = 𝑇)
5149, 50eqeqan12rd 2843 . . . . . . . . . . . . . 14 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑢 = 𝑆𝑤 = 𝑇))
5248, 51anbi12d 624 . . . . . . . . . . . . 13 ((𝑦 = 𝑅𝑢 = 𝑤) → ((𝑦𝐵𝑢 = 𝑆) ↔ (𝑅𝐵𝑤 = 𝑇)))
53 df-mpt 4953 . . . . . . . . . . . . 13 (𝑦𝐵𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦𝐵𝑢 = 𝑆)}
5452, 53brabga 5215 . . . . . . . . . . . 12 ((𝑅𝐵𝑤 ∈ V) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
554, 46, 54sylancl 580 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
56 id 22 . . . . . . . . . . . . 13 (𝑥𝐴𝑥𝐴)
57 eqid 2825 . . . . . . . . . . . . . 14 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
5857fvmpt2 6538 . . . . . . . . . . . . 13 ((𝑥𝐴𝑅𝐵) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
5956, 4, 58syl2an2 677 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
6059breq1d 4883 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑅(𝑦𝐵𝑆)𝑤))
614biantrurd 528 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑤 = 𝑇 ↔ (𝑅𝐵𝑤 = 𝑇)))
6255, 60, 613bitr4d 303 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇))
6362expcom 404 . . . . . . . . 9 (𝑥𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)))
6430, 39, 45, 63vtoclgaf 3488 . . . . . . . 8 (𝑧𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
6564impcom 398 . . . . . . 7 ((𝜑𝑧𝐴) → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
6665pm5.32da 574 . . . . . 6 (𝜑 → ((𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
6729, 66bitrd 271 . . . . 5 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
6819, 67syl5bb 275 . . . 4 (𝜑 → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
6914, 68bitrd 271 . . 3 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
70 vex 3417 . . . 4 𝑧 ∈ V
7170, 46opelco 5526 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤))
72 df-mpt 4953 . . . . 5 (𝑥𝐴𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)}
7372eleq2i 2898 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)})
74 nfv 2013 . . . . . 6 𝑥 𝑧𝐴
7536nfeq2 2985 . . . . . 6 𝑥 𝑣 = 𝑧 / 𝑥𝑇
7674, 75nfan 2002 . . . . 5 𝑥(𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)
77 nfv 2013 . . . . 5 𝑣(𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)
78 eleq1w 2889 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7942eqeq2d 2835 . . . . . 6 (𝑥 = 𝑧 → (𝑣 = 𝑇𝑣 = 𝑧 / 𝑥𝑇))
8078, 79anbi12d 624 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑣 = 𝑇) ↔ (𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)))
81 eqeq1 2829 . . . . . 6 (𝑣 = 𝑤 → (𝑣 = 𝑧 / 𝑥𝑇𝑤 = 𝑧 / 𝑥𝑇))
8281anbi2d 622 . . . . 5 (𝑣 = 𝑤 → ((𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8376, 77, 70, 46, 80, 82opelopabf 5226 . . . 4 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)} ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
8473, 83bitri 267 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
8569, 71, 843bitr4g 306 . 2 (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇)))
861, 2, 85eqrelrdv 5450 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164  Vcvv 3414  ⦋csb 3757  ⟨cop 4403   class class class wbr 4873  {copab 4935   ↦ cmpt 4952  dom cdm 5342   ∘ ccom 5346  Fun wfun 6117  ‘cfv 6123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131 This theorem is referenced by:  fmptcof  6647  cofmpt  6649  fcompt  6650  fcoconst  6651  ofco  7177  ccatco  13956  lo1o12  14641  rlimcn1  14696  rlimcn1b  14697  rlimdiv  14753  ackbijnn  14934  setcepi  17090  prf1st  17197  prf2nd  17198  hofcllem  17251  prdsidlem  17675  pws0g  17679  pwsco1mhm  17723  pwsco2mhm  17724  pwsinvg  17882  pwssub  17883  galactghm  18173  efginvrel1  18492  frgpup3lem  18543  gsumzf1o  18666  gsumconst  18687  gsummptshft  18689  gsumzmhm  18690  gsummhm2  18692  gsummptmhm  18693  gsumsub  18701  gsum2dlem2  18723  dprdfsub  18774  lmhmvsca  19404  psrass1lem  19738  psrlinv  19758  psrcom  19770  evlslem2  19872  coe1fval3  19938  psropprmul  19968  coe1z  19993  coe1mul2  19999  coe1tm  20003  ply1coe  20026  evls1sca  20048  frgpcyg  20281  evpmodpmf1o  20302  mhmvlin  20570  ofco2  20625  mdetleib2  20762  mdetralt  20782  smadiadetlem3  20843  ptrescn  21813  lmcn2  21823  qtopeu  21890  flfcnp2  22181  tgpconncomp  22286  tsmsmhm  22319  tsmssub  22322  tsmsxplem1  22326  negfcncf  23092  pcopt  23191  pcopt2  23192  pi1xfrcnvlem  23225  ovolctb  23656  ovolfs2  23737  uniioombllem2  23749  uniioombllem3  23751  ismbf  23794  mbfconst  23799  ismbfcn2  23804  itg1climres  23880  iblabslem  23993  iblabs  23994  bddmulibl  24004  limccnp  24054  limccnp2  24055  limcco  24056  dvcof  24110  dvcjbr  24111  dvcj  24112  dvfre  24113  dvmptcj  24130  dvmptco  24134  dvcnvlem  24138  dvef  24142  dvlip  24155  dvlipcn  24156  itgsubstlem  24210  plypf1  24367  plyco  24396  dgrcolem1  24428  dgrcolem2  24429  dgrco  24430  plycjlem  24431  taylply2  24521  logcn  24792  leibpi  25082  efrlim  25109  jensenlem2  25127  amgmlem  25129  lgamgulmlem2  25169  lgamcvg2  25194  ftalem7  25218  lgseisenlem4  25516  dchrisum0  25622  ofcfval4  30701  eulerpartgbij  30968  dstfrvclim1  31074  cvmliftlem6  31807  cvmliftphtlem  31834  cvmlift3lem5  31840  elmsubrn  31960  msubco  31963  circum  32101  mblfinlem2  33984  volsupnfl  33991  itgaddnc  34006  itgmulc2nc  34014  ftc1anclem1  34021  ftc1anclem2  34022  ftc1anclem3  34023  ftc1anclem4  34024  ftc1anclem5  34025  ftc1anclem7  34027  ftc1anclem8  34028  fnopabco  34053  upixp  34060  mendassa  38600  fsovrfovd  39136  fsovcnvlem  39140  cncfcompt  40884  dvcosax  40929  dirkercncflem4  41110  fourierdlem111  41221  meadjiunlem  41466  meadjiun  41467  amgmwlem  43437  amgmlemALT  43438
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