1stf2 - Metamath Proof Explorer

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Theorem 1stf2 18184

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)
1stfval.p	⊢ 𝑃 = (𝐶 1^st_F 𝐷)
1stf1.p	⊢ (𝜑 → 𝑅 ∈ 𝐵)
1stf2.p	⊢ (𝜑 → 𝑆 ∈ 𝐵)

**Assertion**
Ref	Expression
1stf2	⊢ (𝜑 → (𝑅(2^nd ‘𝑃)𝑆) = (1^st ↾ (𝑅𝐻𝑆)))

Proof of Theorem 1stf2 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		1stfval.p	. . . 4 ⊢ 𝑃 = (𝐶 1^st_F 𝐷)
7	1, 2, 3, 4, 5, 6	1stfval 18182	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩)
8		fo1st 8013	. . . . . 6 ⊢ 1^st :V–onto→V
9		fofun 6812	. . . . . 6 ⊢ (1^st :V–onto→V → Fun 1^st )
10	8, 9	ax-mp 5	. . . . 5 ⊢ Fun 1^st
11	2	fvexi 6911	. . . . 5 ⊢ 𝐵 ∈ V
12		resfunexg 7227	. . . . 5 ⊢ ((Fun 1^st ∧ 𝐵 ∈ V) → (1^st ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 691	. . . 4 ⊢ (1^st ↾ 𝐵) ∈ V
14	11, 11	mpoex 8084	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op2ndd 8004	. . 3 ⊢ (𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩ → (2^nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))))
16	7, 15	syl 17	. 2 ⊢ (𝜑 → (2^nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))))
17		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅)
18		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆)
19	17, 18	oveq12d 7438	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
20	19	reseq2d 5985	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (1^st ↾ (𝑥𝐻𝑦)) = (1^st ↾ (𝑅𝐻𝑆)))
21		1stf1.p	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
22		1stf2.p	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23		ovex 7453	. . . 4 ⊢ (𝑅𝐻𝑆) ∈ V
24		resfunexg 7227	. . . 4 ⊢ ((Fun 1^st ∧ (𝑅𝐻𝑆) ∈ V) → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
25	10, 23, 24	mp2an 691	. . 3 ⊢ (1^st ↾ (𝑅𝐻𝑆)) ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
27	16, 20, 21, 22, 26	ovmpod 7573	1 ⊢ (𝜑 → (𝑅(2^nd ‘𝑃)𝑆) = (1^st ↾ (𝑅𝐻𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⟨cop 4635 ↾ cres 5680 Fun wfun 6542 –onto→wfo 6546 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7991 2^nd c2nd 7992 Basecbs 17180 Hom chom 17244 Catccat 17644 ×_c cxpc 18159 1^st_F c1stf 18160

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-slot 17151 df-ndx 17163 df-base 17181 df-hom 17257 df-cco 17258 df-xpc 18163 df-1stf 18164

This theorem is referenced by: 1stfcl 18188 prf1st 18195 1st2ndprf 18197 uncf2 18229 diag12 18236 diag2 18237 yonedalem22 18270

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