Theorem 2ndf2 18144

Description: Value of the first projection on a morphism. (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)
2ndf2.p	⊢ (𝜑 → 𝑆 ∈ 𝐵)

Assertion

Ref	Expression
2ndf2	⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))

Proof of Theorem 2ndf2

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

Step	Hyp	Ref	Expression
1		1stfval.t	. . . 4 ⊢ 𝑇 = (𝐶 ×c 𝐷)
2		1stfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑇)
3		1stfval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝑇)
4		1stfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		1stfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		2ndfval.p	. . . 4 ⊢ 𝑄 = (𝐶 2ndF 𝐷)
7	1, 2, 3, 4, 5, 6	2ndfval 18142	. . 3 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8		fo2nd 7992	. . . . . 6 ⊢ 2nd :V–onto→V
9		fofun 6803	. . . . . 6 ⊢ (2nd :V–onto→V → Fun 2nd )
10	8, 9	ax-mp 5	. . . . 5 ⊢ Fun 2nd
11	2	fvexi 6902	. . . . 5 ⊢ 𝐵 ∈ V
12		resfunexg 7213	. . . . 5 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 690	. . . 4 ⊢ (2nd ↾ 𝐵) ∈ V
14	11, 11	mpoex 8062	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op2ndd 7982	. . 3 ⊢ (𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (2nd ‘𝑄) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
16	7, 15	syl 17	. 2 ⊢ (𝜑 → (2nd ‘𝑄) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
17		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅)
18		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆)
19	17, 18	oveq12d 7423	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
20	19	reseq2d 5979	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (2nd ↾ (𝑥𝐻𝑦)) = (2nd ↾ (𝑅𝐻𝑆)))
21		2ndf1.p	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
22		2ndf2.p	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23		ovex 7438	. . . 4 ⊢ (𝑅𝐻𝑆) ∈ V
24		resfunexg 7213	. . . 4 ⊢ ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
25	10, 23, 24	mp2an 690	. . 3 ⊢ (2nd ↾ (𝑅𝐻𝑆)) ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
27	16, 20, 21, 22, 26	ovmpod 7556	1 ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⟨cop 4633   ↾ cres 5677  Fun wfun 6534  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  2^nd c2nd 7970  Basecbs 17140  Hom chom 17204  Catccat 17604   ×_c cxpc 18116   2^nd_F c2ndf 18118

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-slot 17111  df-ndx 17123  df-base 17141  df-hom 17217  df-cco 17218  df-xpc 18120  df-2ndf 18122

This theorem is referenced by:  2ndfcl  18146  prf2nd  18153  1st2ndprf  18154  uncf2  18186  curf2ndf  18196  yonedalem22  18227