1stf1 - Metamath Proof Explorer

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Theorem 1stf1 18140

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)
1stfval.p	⊢ 𝑃 = (𝐶 1^st_F 𝐷)
1stf1.p	⊢ (𝜑 → 𝑅 ∈ 𝐵)

**Assertion**
Ref	Expression
1stf1	⊢ (𝜑 → ((1^st ‘𝑃)‘𝑅) = (1^st ‘𝑅))

Proof of Theorem 1stf1 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		1stfval.p	. . . . 5 ⊢ 𝑃 = (𝐶 1^st_F 𝐷)
7	1, 2, 3, 4, 5, 6	1stfval 18139	. . . 4 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩)
8		fo1st 7991	. . . . . . 7 ⊢ 1^st :V–onto→V
9		fofun 6803	. . . . . . 7 ⊢ (1^st :V–onto→V → Fun 1^st )
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun 1^st
11	2	fvexi 6902	. . . . . 6 ⊢ 𝐵 ∈ V
12		resfunexg 7213	. . . . . 6 ⊢ ((Fun 1^st ∧ 𝐵 ∈ V) → (1^st ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 690	. . . . 5 ⊢ (1^st ↾ 𝐵) ∈ V
14	11, 11	mpoex 8062	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op1std 7981	. . . 4 ⊢ (𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩ → (1^st ‘𝑃) = (1^st ↾ 𝐵))
16	7, 15	syl 17	. . 3 ⊢ (𝜑 → (1^st ‘𝑃) = (1^st ↾ 𝐵))
17	16	fveq1d 6890	. 2 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑅) = ((1^st ↾ 𝐵)‘𝑅))
18		1stf1.p	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
19	18	fvresd 6908	. 2 ⊢ (𝜑 → ((1^st ↾ 𝐵)‘𝑅) = (1^st ‘𝑅))
20	17, 19	eqtrd 2772	1 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑅) = (1^st ‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = 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 1^st c1st 7969 Basecbs 17140 Hom chom 17204 Catccat 17604 ×_c cxpc 18116 1^st_F c1stf 18117

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-1stf 18121

This theorem is referenced by: prf1st 18152 1st2ndprf 18154 uncf1 18185 uncf2 18186 diag11 18192 yonedalem21 18222 yonedalem22 18227

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