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Theorem setcco 16556
 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 16555 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
5 simprr 792 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
76fveq2d 6107 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8 setcco.x . . . . . . . 8 (𝜑𝑋𝑈)
9 setcco.y . . . . . . . 8 (𝜑𝑌𝑈)
10 op2ndg 7072 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
118, 9, 10syl2anc 691 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1211adantr 480 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
137, 12eqtrd 2644 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
145, 13oveq12d 6567 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧𝑚 (2nd𝑣)) = (𝑍𝑚 𝑌))
156fveq2d 6107 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16 op1stg 7071 . . . . . . . 8 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
178, 9, 16syl2anc 691 . . . . . . 7 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1817adantr 480 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1915, 18eqtrd 2644 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = 𝑋)
2013, 19oveq12d 6567 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑣) ↑𝑚 (1st𝑣)) = (𝑌𝑚 𝑋))
21 eqidd 2611 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2214, 20, 21mpt2eq123dv 6615 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)))
23 opelxpi 5072 . . . 4 ((𝑋𝑈𝑌𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
248, 9, 23syl2anc 691 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
25 setcco.z . . 3 (𝜑𝑍𝑈)
26 ovex 6577 . . . . 5 (𝑍𝑚 𝑌) ∈ V
27 ovex 6577 . . . . 5 (𝑌𝑚 𝑋) ∈ V
2826, 27mpt2ex 7136 . . . 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 6686 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ (𝑍𝑚 𝑌), 𝑓 ∈ (𝑌𝑚 𝑋) ↦ (𝑔𝑓)))
31 simprl 790 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
32 simprr 792 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
3331, 32coeq12d 5208 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
34 setcco.g . . 3 (𝜑𝐺:𝑌𝑍)
3525, 9elmapd 7758 . . 3 (𝜑 → (𝐺 ∈ (𝑍𝑚 𝑌) ↔ 𝐺:𝑌𝑍))
3634, 35mpbird 246 . 2 (𝜑𝐺 ∈ (𝑍𝑚 𝑌))
37 setcco.f . . 3 (𝜑𝐹:𝑋𝑌)
389, 8elmapd 7758 . . 3 (𝜑 → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
3937, 38mpbird 246 . 2 (𝜑𝐹 ∈ (𝑌𝑚 𝑋))
40 coexg 7010 . . 3 ((𝐺 ∈ (𝑍𝑚 𝑌) ∧ 𝐹 ∈ (𝑌𝑚 𝑋)) → (𝐺𝐹) ∈ V)
4136, 39, 40syl2anc 691 . 2 (𝜑 → (𝐺𝐹) ∈ V)
4230, 33, 36, 39, 41ovmpt2d 6686 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  compcco 15780  SetCatcsetc 16548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-setc 16549 This theorem is referenced by:  setccatid  16557  setcmon  16560  setcepi  16561  setcsect  16562  resssetc  16565  funcestrcsetclem9  16611  funcsetcestrclem9  16626  hofcllem  16721  yonedalem4c  16740  yonedalem3b  16742  yonedainv  16744  funcringcsetcALTV2lem9  41836  funcringcsetclem9ALTV  41859
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