1stf1 - Metamath Proof Explorer

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Theorem 1stf1 18224

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 18223	. . . 4 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩)
8		fo1st 7990	. . . . . . 7 ⊢ 1^st :V–onto→V
9		fofun 6779	. . . . . . 7 ⊢ (1^st :V–onto→V → Fun 1^st )
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun 1^st
11	2	fvexi 6881	. . . . . 6 ⊢ 𝐵 ∈ V
12		resfunexg 7199	. . . . . 6 ⊢ ((Fun 1^st ∧ 𝐵 ∈ V) → (1^st ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 702	. . . . 5 ⊢ (1^st ↾ 𝐵) ∈ V
14	11, 11	mpoex 8060	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op1std 7980	. . . 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 6869	. 2 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑅) = ((1^st ↾ 𝐵)‘𝑅))
18		1stf1.p	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
19	18	fvresd 6887	. 2 ⊢ (𝜑 → ((1^st ↾ 𝐵)‘𝑅) = (1^st ‘𝑅))
20	17, 19	eqtrd 2797	1 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑅) = (1^st ‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1560 ∈ wcel 2142 Vcvv 3454 ⟨cop 4588 ↾ cres 5649 Fun wfun 6515 –onto→wfo 6519 ‘cfv 6521 (class class class)co 7396 ∈ cmpo 7398 1^st c1st 7968 Basecbs 17245 Hom chom 17297 Catccat 17696 ×_c cxpc 18200 1^st_F c1stf 18201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-slot 17218 df-ndx 17230 df-base 17246 df-hom 17310 df-cco 17311 df-xpc 18204 df-1stf 18205

This theorem is referenced by: prf1st 18236 1st2ndprf 18238 uncf1 18268 uncf2 18269 diag11 18275 yonedalem21 18305 yonedalem22 18310

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