Theorem 2ndf1 17440

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 17439	. . . 4 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8		fo2nd 7703	. . . . . . 7 ⊢ 2nd :V–onto→V
9		fofun 6584	. . . . . . 7 ⊢ (2nd :V–onto→V → Fun 2nd )
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun 2nd
11	2	fvexi 6677	. . . . . 6 ⊢ 𝐵 ∈ V
12		resfunexg 6971	. . . . . 6 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 690	. . . . 5 ⊢ (2nd ↾ 𝐵) ∈ V
14	11, 11	mpoex 7770	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op1std 7692	. . . 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 6665	. 2 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = ((2nd ↾ 𝐵)‘𝑅))
18		2ndf1.p	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
19	18	fvresd 6683	. 2 ⊢ (𝜑 → ((2nd ↾ 𝐵)‘𝑅) = (2nd ‘𝑅))
20	17, 19	eqtrd 2855	1 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  Vcvv 3491  ⟨cop 4566   ↾ cres 5550  Fun wfun 6342  –onto→wfo 6346  ‘cfv 6348  (class class class)co 7149   ∈ cmpo 7151  1^st c1st 7680  2^nd c2nd 7681  Basecbs 16478  Hom chom 16571  Catccat 16930   ×_c cxpc 17413   2^nd_F c2ndf 17415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-er 8282  df-en 8503  df-dom 8504  df-sdom 8505  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11632  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-xpc 17417  df-2ndf 17419

This theorem is referenced by:  prf2nd  17450  1st2ndprf  17451  uncf1  17481  uncf2  17482  curf2ndf  17492  yonedalem21  17518  yonedalem22  17523