Theorem 2ndf1 18152

Description: Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
1stfval.t	⊢ 𝑇 = (𝐶 ×c 𝐷)
1stfval.b	⊢ 𝐵 = (Base‘𝑇)
1stfval.h	⊢ 𝐻 = (Hom ‘𝑇)
1stfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
1stfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
2ndfval.p	⊢ 𝑄 = (𝐶 2ndF 𝐷)
2ndf1.p	⊢ (𝜑 → 𝑅 ∈ 𝐵)

Assertion

Ref	Expression
2ndf1	⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅))

Proof of Theorem 2ndf1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stfval.t	. . . . 5 ⊢ 𝑇 = (𝐶 ×c 𝐷)
2		1stfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑇)
3		1stfval.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝑇)
4		1stfval.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		1stfval.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		2ndfval.p	. . . . 5 ⊢ 𝑄 = (𝐶 2ndF 𝐷)
7	1, 2, 3, 4, 5, 6	2ndfval 18151	. . . 4 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8		fo2nd 7956	. . . . . . 7 ⊢ 2nd :V–onto→V
9		fofun 6747	. . . . . . 7 ⊢ (2nd :V–onto→V → Fun 2nd )
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun 2nd
11	2	fvexi 6848	. . . . . 6 ⊢ 𝐵 ∈ V
12		resfunexg 7163	. . . . . 6 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 693	. . . . 5 ⊢ (2nd ↾ 𝐵) ∈ V
14	11, 11	mpoex 8025	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op1std 7945	. . . 4 ⊢ (𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (1st ‘𝑄) = (2nd ↾ 𝐵))
16	7, 15	syl 17	. . 3 ⊢ (𝜑 → (1st ‘𝑄) = (2nd ↾ 𝐵))
17	16	fveq1d 6836	. 2 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = ((2nd ↾ 𝐵)‘𝑅))
18		2ndf1.p	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
19	18	fvresd 6854	. 2 ⊢ (𝜑 → ((2nd ↾ 𝐵)‘𝑅) = (2nd ‘𝑅))
20	17, 19	eqtrd 2772	1 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   ↾ cres 5626  Fun wfun 6486  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  Hom chom 17222  Catccat 17621   ×_c cxpc 18125   2^nd_F c2ndf 18127

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-xpc 18129  df-2ndf 18131

This theorem is referenced by:  prf2nd  18162  1st2ndprf  18163  uncf1  18193  uncf2  18194  curf2ndf  18204  yonedalem21  18230  yonedalem22  18235