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Theorem setcco 16957
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 16956 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
5 simprr 780 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6 simprl 778 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
76fveq2d 6422 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8 setcco.x . . . . . . . 8 (𝜑𝑋𝑈)
9 setcco.y . . . . . . . 8 (𝜑𝑌𝑈)
10 op2ndg 7421 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
118, 9, 10syl2anc 575 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1211adantr 468 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
137, 12eqtrd 2851 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
145, 13oveq12d 6902 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧𝑚 (2nd𝑣)) = (𝑍𝑚 𝑌))
156fveq2d 6422 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16 op1stg 7420 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
178, 9, 16syl2anc 575 . . . . . . 7 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1817adantr 468 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1915, 18eqtrd 2851 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = 𝑋)
2013, 19oveq12d 6902 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑣) ↑𝑚 (1st𝑣)) = (𝑌𝑚 𝑋))
21 eqidd 2818 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2214, 20, 21mpt2eq123dv 6957 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)))
23 opelxpi 5360 . . . 4 ((𝑋𝑈𝑌𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
248, 9, 23syl2anc 575 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
25 setcco.z . . 3 (𝜑𝑍𝑈)
26 ovex 6916 . . . . 5 (𝑍𝑚 𝑌) ∈ V
27 ovex 6916 . . . . 5 (𝑌𝑚 𝑋) ∈ V
2826, 27mpt2ex 7490 . . . 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 7028 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)))
31 simprl 778 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
32 simprr 780 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
3331, 32coeq12d 5502 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
34 setcco.g . . 3 (𝜑𝐺:𝑌𝑍)
3525, 9elmapd 8116 . . 3 (𝜑 → (𝐺 ∈ (𝑍𝑚 𝑌) ↔ 𝐺:𝑌𝑍))
3634, 35mpbird 248 . 2 (𝜑𝐺 ∈ (𝑍𝑚 𝑌))
37 setcco.f . . 3 (𝜑𝐹:𝑋𝑌)
389, 8elmapd 8116 . . 3 (𝜑 → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
3937, 38mpbird 248 . 2 (𝜑𝐹 ∈ (𝑌𝑚 𝑋))
40 coexg 7357 . . 3 ((𝐺 ∈ (𝑍𝑚 𝑌) ∧ 𝐹 ∈ (𝑌𝑚 𝑋)) → (𝐺𝐹) ∈ V)
4136, 39, 40syl2anc 575 . 2 (𝜑 → (𝐺𝐹) ∈ V)
4230, 33, 36, 39, 41ovmpt2d 7028 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2157  Vcvv 3402  cop 4387   × cxp 5322  ccom 5328  wf 6107  cfv 6111  (class class class)co 6884  cmpt2 6886  1st c1st 7406  2nd c2nd 7407  𝑚 cmap 8102  compcco 16185  SetCatcsetc 16949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-cnex 10287  ax-resscn 10288  ax-1cn 10289  ax-icn 10290  ax-addcl 10291  ax-addrcl 10292  ax-mulcl 10293  ax-mulrcl 10294  ax-mulcom 10295  ax-addass 10296  ax-mulass 10297  ax-distr 10298  ax-i2m1 10299  ax-1ne0 10300  ax-1rid 10301  ax-rnegex 10302  ax-rrecex 10303  ax-cnre 10304  ax-pre-lttri 10305  ax-pre-lttrn 10306  ax-pre-ltadd 10307  ax-pre-mulgt0 10308
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-1o 7806  df-oadd 7810  df-er 7989  df-map 8104  df-en 8203  df-dom 8204  df-sdom 8205  df-fin 8206  df-pnf 10371  df-mnf 10372  df-xr 10373  df-ltxr 10374  df-le 10375  df-sub 10563  df-neg 10564  df-nn 11316  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11580  df-z 11664  df-dec 11780  df-uz 11925  df-fz 12570  df-struct 16090  df-ndx 16091  df-slot 16092  df-base 16094  df-hom 16197  df-cco 16198  df-setc 16950
This theorem is referenced by:  setccatid  16958  setcmon  16961  setcepi  16962  setcsect  16963  resssetc  16966  funcestrcsetclem9  17013  funcsetcestrclem9  17028  hofcllem  17123  yonedalem4c  17142  yonedalem3b  17144  yonedainv  17146  funcringcsetcALTV2lem9  42630  funcringcsetclem9ALTV  42653
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