1stf2 - Metamath Proof Explorer

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Theorem 1stf2 18276

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 18274	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩)
8		fo1st 8013	. . . . . 6 ⊢ 1^st :V–onto→V
9		fofun 6798	. . . . . 6 ⊢ (1^st :V–onto→V → Fun 1^st )
10	8, 9	ax-mp 5	. . . . 5 ⊢ Fun 1^st
11	2	fvexi 6900	. . . . 5 ⊢ 𝐵 ∈ V
12		resfunexg 7221	. . . . 5 ⊢ ((Fun 1^st ∧ 𝐵 ∈ V) → (1^st ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 705	. . . 4 ⊢ (1^st ↾ 𝐵) ∈ V
14	11, 11	mpoex 8083	. . . 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 18	. 2 ⊢ (𝜑 → (2^nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (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 7438	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
20	19	reseq2d 5980	. 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 7221	. . . 4 ⊢ ((Fun 1^st ∧ (𝑅𝐻𝑆) ∈ V) → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
25	10, 23, 24	mp2an 705	. . 3 ⊢ (1^st ↾ (𝑅𝐻𝑆)) ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
27	16, 20, 21, 22, 26	ovmpod 7572	1 ⊢ (𝜑 → (𝑅(2^nd ‘𝑃)𝑆) = (1^st ↾ (𝑅𝐻𝑆)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3457 ⟨cop 4597 ↾ cres 5665 Fun wfun 6535 –onto→wfo 6539 ‘cfv 6541 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7991 2^nd c2nd 7992 Basecbs 17296 Hom chom 17348 Catccat 17747 ×_c cxpc 18251 1^st_F c1stf 18252

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7743 ax-cnex 11176 ax-resscn 11177 ax-1cn 11178 ax-icn 11179 ax-addcl 11180 ax-addrcl 11181 ax-mulcl 11182 ax-mulrcl 11183 ax-mulcom 11184 ax-addass 11185 ax-mulass 11186 ax-distr 11187 ax-i2m1 11188 ax-1ne0 11189 ax-1rid 11190 ax-rnegex 11191 ax-rrecex 11192 ax-cnre 11193 ax-pre-lttri 11194 ax-pre-lttrn 11195 ax-pre-ltadd 11196 ax-pre-mulgt0 11197

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7870 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-er 8701 df-en 8951 df-dom 8952 df-sdom 8953 df-pnf 11265 df-mnf 11266 df-xr 11267 df-ltxr 11268 df-le 11269 df-sub 11463 df-neg 11464 df-nn 12254 df-2 12323 df-3 12324 df-4 12325 df-5 12326 df-6 12327 df-7 12328 df-8 12329 df-9 12330 df-n0 12525 df-z 12612 df-dec 12733 df-slot 17269 df-ndx 17281 df-base 17297 df-hom 17361 df-cco 17362 df-xpc 18255 df-1stf 18256

This theorem is used by: 1stfcl 18280 prf1st 18287 1st2ndprf 18289 uncf2 18320 diag12 18327 diag2 18328 yonedalem22 18361

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