Theorem 2ndf2 18399

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 18397	. . 3 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8		fo2nd 8046	. . . . . 6 ⊢ 2nd :V⟶ontoV
9		fofun 6819	. . . . . 6 ⊢ (2nd :V⟶ontoV → Fun 2nd )
10	8, 9	ax-mp 5	. . . . 5 ⊢ Fun 2nd
11	2	fvexi 6922	. . . . 5 ⊢ 𝐵 ∈ V
12		resfunexg 7244	. . . . 5 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 705	. . . 4 ⊢ (2nd ↾ 𝐵) ∈ V
14	11, 11	mpoex 8117	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op2ndd 8036	. . 3 ⊢ (𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (2nd ‘𝑄) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
16	7, 15	syl 18	. 2 ⊢ (𝜑 → (2nd ‘𝑄) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
17		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅)
18		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆)
19	17, 18	oveq12d 7461	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
20	19	reseq2d 5981	. 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 7476	. . . 4 ⊢ (𝑅𝐻𝑆) ∈ V
24		resfunexg 7244	. . . 4 ⊢ ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
25	10, 23, 24	mp2an 705	. . 3 ⊢ (2nd ↾ (𝑅𝐻𝑆)) ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
27	16, 20, 21, 22, 26	ovmpod 7596	1 ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838   ↾ cres 5657  Fun wfun 6552  ⟶ontowfo 6556  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  2^nd c2nd 8024  Basecbs 17416  Hom chom 17468  Catccat 17867   ×_c cxpc 18371   2^nd_F c2ndf 18373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-xpc 18375  df-2ndf 18377

This theorem is used by:  2ndfcl  18401  prf2nd  18408  1st2ndprf  18409  uncf2  18440  curf2ndf  18450  yonedalem22  18481