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Mirrors > Home > MPE Home > Th. List > 2ndf2 | Structured version Visualization version GIF version |
Description: Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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) |
2ndfval.p | ⊢ 𝑄 = (𝐶 2ndF 𝐷) |
2ndf1.p | ⊢ (𝜑 → 𝑅 ∈ 𝐵) |
2ndf2.p | ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
Ref | Expression |
---|---|
2ndf2 | ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆))) |
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 | 2ndfval.p | . . . 4 ⊢ 𝑄 = (𝐶 2ndF 𝐷) | |
7 | 1, 2, 3, 4, 5, 6 | 2ndfval 18185 | . . 3 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩) |
8 | fo2nd 8014 | . . . . . 6 ⊢ 2nd :V–onto→V | |
9 | fofun 6812 | . . . . . 6 ⊢ (2nd :V–onto→V → Fun 2nd ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ Fun 2nd |
11 | 2 | fvexi 6911 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | resfunexg 7227 | . . . . 5 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V) | |
13 | 10, 11, 12 | mp2an 691 | . . . 4 ⊢ (2nd ↾ 𝐵) ∈ V |
14 | 11, 11 | mpoex 8084 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V |
15 | 13, 14 | op2ndd 8004 | . . 3 ⊢ (𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (2nd ‘𝑄) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))) |
16 | 7, 15 | syl 17 | . 2 ⊢ (𝜑 → (2nd ‘𝑄) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (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)) → (2nd ↾ (𝑥𝐻𝑦)) = (2nd ↾ (𝑅𝐻𝑆))) |
21 | 2ndf1.p | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝐵) | |
22 | 2ndf2.p | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝐵) | |
23 | ovex 7453 | . . . 4 ⊢ (𝑅𝐻𝑆) ∈ V | |
24 | resfunexg 7227 | . . . 4 ⊢ ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V) | |
25 | 10, 23, 24 | mp2an 691 | . . 3 ⊢ (2nd ↾ (𝑅𝐻𝑆)) ∈ V |
26 | 25 | a1i 11 | . 2 ⊢ (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V) |
27 | 16, 20, 21, 22, 26 | ovmpod 7573 | 1 ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (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 2nd c2nd 7992 Basecbs 17180 Hom chom 17244 Catccat 17644 ×c cxpc 18159 2ndF c2ndf 18161 |
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-2ndf 18165 |
This theorem is referenced by: 2ndfcl 18189 prf2nd 18196 1st2ndprf 18197 uncf2 18229 curf2ndf 18239 yonedalem22 18270 |
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