1stf2 - Metamath Proof Explorer

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Theorem 1stf2 18226

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 18224	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩)
8		fo1st 7991	. . . . . 6 ⊢ 1^st :V–onto→V
9		fofun 6780	. . . . . 6 ⊢ (1^st :V–onto→V → Fun 1^st )
10	8, 9	ax-mp 5	. . . . 5 ⊢ Fun 1^st
11	2	fvexi 6882	. . . . 5 ⊢ 𝐵 ∈ V
12		resfunexg 7200	. . . . 5 ⊢ ((Fun 1^st ∧ 𝐵 ∈ V) → (1^st ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 702	. . . 4 ⊢ (1^st ↾ 𝐵) ∈ V
14	11, 11	mpoex 8061	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op2ndd 7982	. . 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 780	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅)
18		simprr 782	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆)
19	17, 18	oveq12d 7415	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
20	19	reseq2d 5966	. 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 7430	. . . 4 ⊢ (𝑅𝐻𝑆) ∈ V
24		resfunexg 7200	. . . 4 ⊢ ((Fun 1^st ∧ (𝑅𝐻𝑆) ∈ V) → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
25	10, 23, 24	mp2an 702	. . 3 ⊢ (1^st ↾ (𝑅𝐻𝑆)) ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
27	16, 20, 21, 22, 26	ovmpod 7549	1 ⊢ (𝜑 → (𝑅(2^nd ‘𝑃)𝑆) = (1^st ↾ (𝑅𝐻𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1561 ∈ wcel 2143 Vcvv 3455 ⟨cop 4589 ↾ cres 5650 Fun wfun 6516 –onto→wfo 6520 ‘cfv 6522 (class class class)co 7397 ∈ cmpo 7399 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17246 Hom chom 17298 Catccat 17697 ×_c cxpc 18201 1^st_F c1stf 18202

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-om 7848 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-er 8679 df-en 8929 df-dom 8930 df-sdom 8931 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-nn 12212 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12483 df-z 12570 df-dec 12690 df-slot 17219 df-ndx 17231 df-base 17247 df-hom 17311 df-cco 17312 df-xpc 18205 df-1stf 18206

This theorem is referenced by: 1stfcl 18230 prf1st 18237 1st2ndprf 18239 uncf2 18270 diag12 18277 diag2 18278 yonedalem22 18311

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