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Theorem setcco 18020
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 18019 . . 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 6885 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8 setcco.x . . . . . . . 8 (𝜑𝑋𝑈)
9 setcco.y . . . . . . . 8 (𝜑𝑌𝑈)
10 op2ndg 7975 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
118, 9, 10syl2anc 585 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1211adantr 482 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
137, 12eqtrd 2773 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
145, 13oveq12d 7414 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧m (2nd𝑣)) = (𝑍m 𝑌))
156fveq2d 6885 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16 op1stg 7974 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
178, 9, 16syl2anc 585 . . . . . . 7 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1817adantr 482 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1915, 18eqtrd 2773 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = 𝑋)
2013, 19oveq12d 7414 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑣) ↑m (1st𝑣)) = (𝑌m 𝑋))
21 eqidd 2734 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2214, 20, 21mpoeq123dv 7471 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑍m 𝑌), 𝑓 ∈ (𝑌m 𝑋) ↦ (𝑔𝑓)))
238, 9opelxpd 5710 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
24 setcco.z . . 3 (𝜑𝑍𝑈)
25 ovex 7429 . . . . 5 (𝑍m 𝑌) ∈ V
26 ovex 7429 . . . . 5 (𝑌m 𝑋) ∈ V
2725, 26mpoex 8053 . . . 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 7547 . 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 5859 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
33 setcco.g . . 3 (𝜑𝐺:𝑌𝑍)
3424, 9elmapd 8822 . . 3 (𝜑 → (𝐺 ∈ (𝑍m 𝑌) ↔ 𝐺:𝑌𝑍))
3533, 34mpbird 257 . 2 (𝜑𝐺 ∈ (𝑍m 𝑌))
36 setcco.f . . 3 (𝜑𝐹:𝑋𝑌)
379, 8elmapd 8822 . . 3 (𝜑 → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
3836, 37mpbird 257 . 2 (𝜑𝐹 ∈ (𝑌m 𝑋))
39 coexg 7907 . . 3 ((𝐺 ∈ (𝑍m 𝑌) ∧ 𝐹 ∈ (𝑌m 𝑋)) → (𝐺𝐹) ∈ V)
4035, 38, 39syl2anc 585 . 2 (𝜑 → (𝐺𝐹) ∈ V)
4129, 32, 35, 38, 40ovmpod 7547 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cop 4630   × cxp 5670  ccom 5676  wf 6531  cfv 6535  (class class class)co 7396  cmpo 7398  1st c1st 7960  2nd c2nd 7961  m cmap 8808  compcco 17196  SetCatcsetc 18012
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 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-3 12263  df-4 12264  df-5 12265  df-6 12266  df-7 12267  df-8 12268  df-9 12269  df-n0 12460  df-z 12546  df-dec 12665  df-uz 12810  df-fz 13472  df-struct 17067  df-slot 17102  df-ndx 17114  df-base 17132  df-hom 17208  df-cco 17209  df-setc 18013
This theorem is referenced by:  setccatid  18021  setcmon  18024  setcepi  18025  setcsect  18026  resssetc  18029  funcestrcsetclem9  18087  funcsetcestrclem9  18102  hofcllem  18198  yonedalem4c  18217  yonedalem3b  18219  yonedainv  18221  funcringcsetcALTV2lem9  46782  funcringcsetclem9ALTV  46805
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