Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 17444 | . . 3 ⊢ (𝜑 → 𝑄 = 〈(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))〉) |
8 | fo2nd 7710 | . . . . . 6 ⊢ 2nd :V–onto→V | |
9 | fofun 6591 | . . . . . 6 ⊢ (2nd :V–onto→V → Fun 2nd ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ Fun 2nd |
11 | 2 | fvexi 6684 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | resfunexg 6978 | . . . . 5 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V) | |
13 | 10, 11, 12 | mp2an 690 | . . . 4 ⊢ (2nd ↾ 𝐵) ∈ V |
14 | 11, 11 | mpoex 7777 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V |
15 | 13, 14 | op2ndd 7700 | . . 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 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
18 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
19 | 17, 18 | oveq12d 7174 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆)) |
20 | 19 | reseq2d 5853 | . 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 7189 | . . . 4 ⊢ (𝑅𝐻𝑆) ∈ V | |
24 | resfunexg 6978 | . . . 4 ⊢ ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V) | |
25 | 10, 23, 24 | mp2an 690 | . . 3 ⊢ (2nd ↾ (𝑅𝐻𝑆)) ∈ V |
26 | 25 | a1i 11 | . 2 ⊢ (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V) |
27 | 16, 20, 21, 22, 26 | ovmpod 7302 | 1 ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 ↾ cres 5557 Fun wfun 6349 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 2nd c2nd 7688 Basecbs 16483 Hom chom 16576 Catccat 16935 ×c cxpc 17418 2ndF c2ndf 17420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-ndx 16486 df-slot 16487 df-base 16489 df-hom 16589 df-cco 16590 df-xpc 17422 df-2ndf 17424 |
This theorem is referenced by: 2ndfcl 17448 prf2nd 17455 1st2ndprf 17456 uncf2 17487 curf2ndf 17497 yonedalem22 17528 |
Copyright terms: Public domain | W3C validator |