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Theorem setcco 16954
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 16953 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
5 simprr 813 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
76fveq2d 6357 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8 setcco.x . . . . . . . 8 (𝜑𝑋𝑈)
9 setcco.y . . . . . . . 8 (𝜑𝑌𝑈)
10 op2ndg 7347 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
118, 9, 10syl2anc 696 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1211adantr 472 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
137, 12eqtrd 2794 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
145, 13oveq12d 6832 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧𝑚 (2nd𝑣)) = (𝑍𝑚 𝑌))
156fveq2d 6357 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16 op1stg 7346 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
178, 9, 16syl2anc 696 . . . . . . 7 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1817adantr 472 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1915, 18eqtrd 2794 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = 𝑋)
2013, 19oveq12d 6832 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑣) ↑𝑚 (1st𝑣)) = (𝑌𝑚 𝑋))
21 eqidd 2761 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2214, 20, 21mpt2eq123dv 6883 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)))
23 opelxpi 5305 . . . 4 ((𝑋𝑈𝑌𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
248, 9, 23syl2anc 696 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
25 setcco.z . . 3 (𝜑𝑍𝑈)
26 ovex 6842 . . . . 5 (𝑍𝑚 𝑌) ∈ V
27 ovex 6842 . . . . 5 (𝑌𝑚 𝑋) ∈ V
2826, 27mpt2ex 7416 . . . 4 (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)) ∈ V
2928a1i 11 . . 3 (𝜑 → (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)) ∈ V)
304, 22, 24, 25, 29ovmpt2d 6954 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)))
31 simprl 811 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
32 simprr 813 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
3331, 32coeq12d 5442 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
34 setcco.g . . 3 (𝜑𝐺:𝑌𝑍)
3525, 9elmapd 8039 . . 3 (𝜑 → (𝐺 ∈ (𝑍𝑚 𝑌) ↔ 𝐺:𝑌𝑍))
3634, 35mpbird 247 . 2 (𝜑𝐺 ∈ (𝑍𝑚 𝑌))
37 setcco.f . . 3 (𝜑𝐹:𝑋𝑌)
389, 8elmapd 8039 . . 3 (𝜑 → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
3937, 38mpbird 247 . 2 (𝜑𝐹 ∈ (𝑌𝑚 𝑋))
40 coexg 7283 . . 3 ((𝐺 ∈ (𝑍𝑚 𝑌) ∧ 𝐹 ∈ (𝑌𝑚 𝑋)) → (𝐺𝐹) ∈ V)
4136, 39, 40syl2anc 696 . 2 (𝜑 → (𝐺𝐹) ∈ V)
4230, 33, 36, 39, 41ovmpt2d 6954 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327   × cxp 5264  ccom 5270  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  1st c1st 7332  2nd c2nd 7333  𝑚 cmap 8025  compcco 16175  SetCatcsetc 16946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-setc 16947
This theorem is referenced by:  setccatid  16955  setcmon  16958  setcepi  16959  setcsect  16960  resssetc  16963  funcestrcsetclem9  17009  funcsetcestrclem9  17024  hofcllem  17119  yonedalem4c  17138  yonedalem3b  17140  yonedainv  17142  funcringcsetcALTV2lem9  42572  funcringcsetclem9ALTV  42595
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