1stf1 - Metamath Proof Explorer

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Theorem 1stf1 18281

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 18280	. . . 4 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩)
8		fo1st 8007	. . . . . . 7 ⊢ 1^st :V–onto→V
9		fofun 6791	. . . . . . 7 ⊢ (1^st :V–onto→V → Fun 1^st )
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun 1^st
11	2	fvexi 6893	. . . . . 6 ⊢ 𝐵 ∈ V
12		resfunexg 7215	. . . . . 6 ⊢ ((Fun 1^st ∧ 𝐵 ∈ V) → (1^st ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 705	. . . . 5 ⊢ (1^st ↾ 𝐵) ∈ V
14	11, 11	mpoex 8079	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op1std 7997	. . . 4 ⊢ (𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩ → (1^st ‘𝑃) = (1^st ↾ 𝐵))
16	7, 15	syl 18	. . 3 ⊢ (𝜑 → (1^st ‘𝑃) = (1^st ↾ 𝐵))
17	16	fveq1d 6881	. 2 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑅) = ((1^st ↾ 𝐵)‘𝑅))
18		1stf1.p	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
19	18	fvresd 6899	. 2 ⊢ (𝜑 → ((1^st ↾ 𝐵)‘𝑅) = (1^st ‘𝑅))
20	17, 19	eqtrd 2795	1 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑅) = (1^st ‘𝑅))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ⟨cop 4590 ↾ cres 5657 Fun wfun 6527 –onto→wfo 6531 ‘cfv 6533 (class class class)co 7414 ∈ cmpo 7416 1^st c1st 7985 Basecbs 17302 Hom chom 17354 Catccat 17753 ×_c cxpc 18257 1^st_F c1stf 18258

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8697 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-slot 17275 df-ndx 17287 df-base 17303 df-hom 17367 df-cco 17368 df-xpc 18261 df-1stf 18262

This theorem is used by: prf1st 18293 1st2ndprf 18295 uncf1 18325 uncf2 18326 diag11 18332 yonedalem21 18362 yonedalem22 18367

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