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Theorem fmptco 7076
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 (𝐺𝐹) 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 6061 . 2 Rel (𝐺𝐹)
2 mptrel 5782 . 2 Rel (𝑥𝐴𝑇)
3 fmptco.2 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑥𝐴𝑅))
4 fmptco.1 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑅𝐵)
53, 4fmpt3d 7065 . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
65ffund 6673 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
7 funbrfv 6894 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹𝑧) = 𝑢))
87imp 408 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
96, 8sylan 581 . . . . . . . . 9 ((𝜑𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
109eqcomd 2743 . . . . . . . 8 ((𝜑𝑧𝐹𝑢) → 𝑢 = (𝐹𝑧))
1110a1d 25 . . . . . . 7 ((𝜑𝑧𝐹𝑢) → (𝑢𝐺𝑤𝑢 = (𝐹𝑧)))
1211expimpd 455 . . . . . 6 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) → 𝑢 = (𝐹𝑧)))
1312pm4.71rd 564 . . . . 5 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
1413exbidv 1925 . . . 4 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
15 fvex 6856 . . . . . 6 (𝐹𝑧) ∈ V
16 breq2 5110 . . . . . . 7 (𝑢 = (𝐹𝑧) → (𝑧𝐹𝑢𝑧𝐹(𝐹𝑧)))
17 breq1 5109 . . . . . . 7 (𝑢 = (𝐹𝑧) → (𝑢𝐺𝑤 ↔ (𝐹𝑧)𝐺𝑤))
1816, 17anbi12d 632 . . . . . 6 (𝑢 = (𝐹𝑧) → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤)))
1915, 18ceqsexv 3495 . . . . 5 (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤))
20 funfvbrb 7002 . . . . . . . . 9 (Fun 𝐹 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
216, 20syl 17 . . . . . . . 8 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
225fdmd 6680 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐴)
2322eleq2d 2824 . . . . . . . 8 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
2421, 23bitr3d 281 . . . . . . 7 (𝜑 → (𝑧𝐹(𝐹𝑧) ↔ 𝑧𝐴))
253fveq1d 6845 . . . . . . . 8 (𝜑 → (𝐹𝑧) = ((𝑥𝐴𝑅)‘𝑧))
26 fmptco.3 . . . . . . . 8 (𝜑𝐺 = (𝑦𝐵𝑆))
27 eqidd 2738 . . . . . . . 8 (𝜑𝑤 = 𝑤)
2825, 26, 27breq123d 5120 . . . . . . 7 (𝜑 → ((𝐹𝑧)𝐺𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
2924, 28anbi12d 632 . . . . . 6 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤)))
30 nfcv 2908 . . . . . . . . 9 𝑥𝑧
31 nfv 1918 . . . . . . . . . 10 𝑥𝜑
32 nffvmpt1 6854 . . . . . . . . . . . 12 𝑥((𝑥𝐴𝑅)‘𝑧)
33 nfcv 2908 . . . . . . . . . . . 12 𝑥(𝑦𝐵𝑆)
34 nfcv 2908 . . . . . . . . . . . 12 𝑥𝑤
3532, 33, 34nfbr 5153 . . . . . . . . . . 11 𝑥((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤
36 nfcsb1v 3881 . . . . . . . . . . . 12 𝑥𝑧 / 𝑥𝑇
3736nfeq2 2925 . . . . . . . . . . 11 𝑥 𝑤 = 𝑧 / 𝑥𝑇
3835, 37nfbi 1907 . . . . . . . . . 10 𝑥(((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)
3931, 38nfim 1900 . . . . . . . . 9 𝑥(𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
40 fveq2 6843 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐴𝑅)‘𝑥) = ((𝑥𝐴𝑅)‘𝑧))
4140breq1d 5116 . . . . . . . . . . 11 (𝑥 = 𝑧 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
42 csbeq1a 3870 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑇 = 𝑧 / 𝑥𝑇)
4342eqeq2d 2748 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑤 = 𝑇𝑤 = 𝑧 / 𝑥𝑇))
4441, 43bibi12d 346 . . . . . . . . . 10 (𝑥 = 𝑧 → ((((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇) ↔ (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
4544imbi2d 341 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)) ↔ (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))))
46 vex 3450 . . . . . . . . . . . 12 𝑤 ∈ V
47 simpl 484 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑦 = 𝑅)
4847eleq1d 2823 . . . . . . . . . . . . . 14 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑦𝐵𝑅𝐵))
49 id 22 . . . . . . . . . . . . . . 15 (𝑢 = 𝑤𝑢 = 𝑤)
50 fmptco.4 . . . . . . . . . . . . . . 15 (𝑦 = 𝑅𝑆 = 𝑇)
5149, 50eqeqan12rd 2752 . . . . . . . . . . . . . 14 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑢 = 𝑆𝑤 = 𝑇))
5248, 51anbi12d 632 . . . . . . . . . . . . 13 ((𝑦 = 𝑅𝑢 = 𝑤) → ((𝑦𝐵𝑢 = 𝑆) ↔ (𝑅𝐵𝑤 = 𝑇)))
53 df-mpt 5190 . . . . . . . . . . . . 13 (𝑦𝐵𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦𝐵𝑢 = 𝑆)}
5452, 53brabga 5492 . . . . . . . . . . . 12 ((𝑅𝐵𝑤 ∈ V) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
554, 46, 54sylancl 587 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
56 id 22 . . . . . . . . . . . . 13 (𝑥𝐴𝑥𝐴)
57 eqid 2737 . . . . . . . . . . . . . 14 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
5857fvmpt2 6960 . . . . . . . . . . . . 13 ((𝑥𝐴𝑅𝐵) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
5956, 4, 58syl2an2 685 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
6059breq1d 5116 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑅(𝑦𝐵𝑆)𝑤))
614biantrurd 534 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑤 = 𝑇 ↔ (𝑅𝐵𝑤 = 𝑇)))
6255, 60, 613bitr4d 311 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇))
6362expcom 415 . . . . . . . . 9 (𝑥𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)))
6430, 39, 45, 63vtoclgaf 3534 . . . . . . . 8 (𝑧𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
6564impcom 409 . . . . . . 7 ((𝜑𝑧𝐴) → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
6665pm5.32da 580 . . . . . 6 (𝜑 → ((𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
6729, 66bitrd 279 . . . . 5 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
6819, 67bitrid 283 . . . 4 (𝜑 → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
6914, 68bitrd 279 . . 3 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
70 vex 3450 . . . 4 𝑧 ∈ V
7170, 46opelco 5828 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤))
72 df-mpt 5190 . . . . 5 (𝑥𝐴𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)}
7372eleq2i 2830 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)})
74 nfv 1918 . . . . . 6 𝑥 𝑧𝐴
7536nfeq2 2925 . . . . . 6 𝑥 𝑣 = 𝑧 / 𝑥𝑇
7674, 75nfan 1903 . . . . 5 𝑥(𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)
77 nfv 1918 . . . . 5 𝑣(𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)
78 eleq1w 2821 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7942eqeq2d 2748 . . . . . 6 (𝑥 = 𝑧 → (𝑣 = 𝑇𝑣 = 𝑧 / 𝑥𝑇))
8078, 79anbi12d 632 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑣 = 𝑇) ↔ (𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)))
81 eqeq1 2741 . . . . . 6 (𝑣 = 𝑤 → (𝑣 = 𝑧 / 𝑥𝑇𝑤 = 𝑧 / 𝑥𝑇))
8281anbi2d 630 . . . . 5 (𝑣 = 𝑤 → ((𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8376, 77, 70, 46, 80, 82opelopabf 5503 . . . 4 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)} ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
8473, 83bitri 275 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
8569, 71, 843bitr4g 314 . 2 (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇)))
861, 2, 85eqrelrdv 5749 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3446  csb 3856  cop 4593   class class class wbr 5106  {copab 5168  cmpt 5189  dom cdm 5634  ccom 5638  Fun wfun 6491  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  fmptcof  7077  cofmpt  7079  fcompt  7080  fcoconst  7081  ofco  7641  ccatco  14725  rlimcn1  15471  rlimdiv  15531  ackbijnn  15714  setcepi  17975  prf1st  18093  prf2nd  18094  hofcllem  18148  prdsidlem  18589  pws0g  18593  pwsco1mhm  18643  pwsco2mhm  18644  smndex1iidm  18712  smndex2dlinvh  18728  pwsinvg  18861  pwssub  18862  galactghm  19187  efginvrel1  19511  frgpup3lem  19560  gsumzf1o  19690  gsumconst  19712  gsummptshft  19714  gsumzmhm  19715  gsummhm2  19717  gsummptmhm  19718  gsumsub  19726  gsum2dlem2  19749  dprdfsub  19801  lmhmvsca  20509  frgpcyg  20983  evpmodpmf1o  21003  psrass1lemOLD  21345  psrass1lem  21348  psrlinv  21368  psrcom  21381  evlslem2  21492  coe1fval3  21582  psropprmul  21612  coe1z  21637  coe1mul2  21643  coe1tm  21647  ply1coe  21670  evls1sca  21692  mhmvlin  21749  ofco2  21803  mdetleib2  21940  mdetralt  21960  smadiadetlem3  22020  ptrescn  22993  lmcn2  23003  qtopeu  23070  flfcnp2  23361  tgpconncomp  23467  tsmssub  23503  tsmsxplem1  23507  negfcncf  24289  pcopt  24388  pcopt2  24389  pi1xfrcnvlem  24422  ovolctb  24857  ovolfs2  24938  uniioombllem2  24950  ismbf  24995  mbfconst  25000  limccnp2  25259  limcco  25260  dvcof  25315  dvcj  25317  dvfre  25318  dvmptcj  25335  dvmptco  25339  dvcnvlem  25343  dvlip  25360  dvlipcn  25361  itgsubstlem  25415  plyco  25605  dgrcolem1  25637  dgrcolem2  25638  dgrco  25639  plycjlem  25640  taylply2  25730  logcn  26005  leibpi  26295  efrlim  26322  jensenlem2  26340  amgmlem  26342  ftalem7  26431  dchrisum0  26871  ghmquskerco  32199  ofcfval4  32707  eulerpartgbij  32975  dstfrvclim1  33080  cvmliftlem6  33887  cvmliftphtlem  33914  cvmlift3lem5  33920  elmsubrn  34125  msubco  34128  circum  34265  mblfinlem2  36119  volsupnfl  36126  itgaddnc  36141  itgmulc2nc  36149  ftc1anclem1  36154  ftc1anclem2  36155  ftc1anclem3  36156  ftc1anclem4  36157  ftc1anclem5  36158  ftc1anclem7  36160  ftc1anclem8  36161  fnopabco  36185  upixp  36191  mendassa  41524  fsovrfovd  42288  fsovcnvlem  42292  cncfcompt  44131  dvcosax  44174  dirkercncflem4  44354  fourierdlem111  44465  meadjiunlem  44713  meadjiun  44714  fundcmpsurbijinjpreimafv  45606  itcovalpclem2  46764  itcovalt2lem2  46769  amgmwlem  47256  amgmlemALT  47257
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