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Theorem setcco 17338
Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcbas.c 𝐶 = (SetCat‘𝑈)
setcbas.u (𝜑𝑈𝑉)
setcco.o · = (comp‘𝐶)
setcco.x (𝜑𝑋𝑈)
setcco.y (𝜑𝑌𝑈)
setcco.z (𝜑𝑍𝑈)
setcco.f (𝜑𝐹:𝑋𝑌)
setcco.g (𝜑𝐺:𝑌𝑍)
Assertion
Ref Expression
setcco (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Proof of Theorem setcco
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 setcbas.c . . . 4 𝐶 = (SetCat‘𝑈)
2 setcbas.u . . . 4 (𝜑𝑈𝑉)
3 setcco.o . . . 4 · = (comp‘𝐶)
41, 2, 3setccofval 17337 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))))
5 simprr 772 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
76fveq2d 6653 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8 setcco.x . . . . . . . 8 (𝜑𝑋𝑈)
9 setcco.y . . . . . . . 8 (𝜑𝑌𝑈)
10 op2ndg 7688 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
118, 9, 10syl2anc 587 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1211adantr 484 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
137, 12eqtrd 2836 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
145, 13oveq12d 7157 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧m (2nd𝑣)) = (𝑍m 𝑌))
156fveq2d 6653 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16 op1stg 7687 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
178, 9, 16syl2anc 587 . . . . . . 7 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1817adantr 484 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1915, 18eqtrd 2836 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = 𝑋)
2013, 19oveq12d 7157 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑣) ↑m (1st𝑣)) = (𝑌m 𝑋))
21 eqidd 2802 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2214, 20, 21mpoeq123dv 7212 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑍m 𝑌), 𝑓 ∈ (𝑌m 𝑋) ↦ (𝑔𝑓)))
238, 9opelxpd 5561 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
24 setcco.z . . 3 (𝜑𝑍𝑈)
25 ovex 7172 . . . . 5 (𝑍m 𝑌) ∈ V
26 ovex 7172 . . . . 5 (𝑌m 𝑋) ∈ V
2725, 26mpoex 7764 . . . 4 (𝑔 ∈ (𝑍m 𝑌), 𝑓 ∈ (𝑌m 𝑋) ↦ (𝑔𝑓)) ∈ V
2827a1i 11 . . 3 (𝜑 → (𝑔 ∈ (𝑍m 𝑌), 𝑓 ∈ (𝑌m 𝑋) ↦ (𝑔𝑓)) ∈ V)
294, 22, 23, 24, 28ovmpod 7285 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ (𝑍m 𝑌), 𝑓 ∈ (𝑌m 𝑋) ↦ (𝑔𝑓)))
30 simprl 770 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
31 simprr 772 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
3230, 31coeq12d 5703 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
33 setcco.g . . 3 (𝜑𝐺:𝑌𝑍)
3424, 9elmapd 8407 . . 3 (𝜑 → (𝐺 ∈ (𝑍m 𝑌) ↔ 𝐺:𝑌𝑍))
3533, 34mpbird 260 . 2 (𝜑𝐺 ∈ (𝑍m 𝑌))
36 setcco.f . . 3 (𝜑𝐹:𝑋𝑌)
379, 8elmapd 8407 . . 3 (𝜑 → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
3836, 37mpbird 260 . 2 (𝜑𝐹 ∈ (𝑌m 𝑋))
39 coexg 7620 . . 3 ((𝐺 ∈ (𝑍m 𝑌) ∧ 𝐹 ∈ (𝑌m 𝑋)) → (𝐺𝐹) ∈ V)
4035, 38, 39syl2anc 587 . 2 (𝜑 → (𝐺𝐹) ∈ V)
4129, 32, 35, 38, 40ovmpod 7285 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  cop 4534   × cxp 5521  ccom 5527  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  1st c1st 7673  2nd c2nd 7674  m cmap 8393  compcco 16572  SetCatcsetc 17330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-setc 17331
This theorem is referenced by:  setccatid  17339  setcmon  17342  setcepi  17343  setcsect  17344  resssetc  17347  funcestrcsetclem9  17393  funcsetcestrclem9  17408  hofcllem  17503  yonedalem4c  17522  yonedalem3b  17524  yonedainv  17526  funcringcsetcALTV2lem9  44655  funcringcsetclem9ALTV  44678
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