Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 1stf2 | 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) |
1stfval.p | ⊢ 𝑃 = (𝐶 1stF 𝐷) |
1stf1.p | ⊢ (𝜑 → 𝑅 ∈ 𝐵) |
1stf2.p | ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
Ref | Expression |
---|---|
1stf2 | ⊢ (𝜑 → (𝑅(2nd ‘𝑃)𝑆) = (1st ↾ (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 | 1stfval.p | . . . 4 ⊢ 𝑃 = (𝐶 1stF 𝐷) | |
7 | 1, 2, 3, 4, 5, 6 | 1stfval 17695 | . . 3 ⊢ (𝜑 → 𝑃 = 〈(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))〉) |
8 | fo1st 7778 | . . . . . 6 ⊢ 1st :V–onto→V | |
9 | fofun 6631 | . . . . . 6 ⊢ (1st :V–onto→V → Fun 1st ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ Fun 1st |
11 | 2 | fvexi 6728 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | resfunexg 7028 | . . . . 5 ⊢ ((Fun 1st ∧ 𝐵 ∈ V) → (1st ↾ 𝐵) ∈ V) | |
13 | 10, 11, 12 | mp2an 692 | . . . 4 ⊢ (1st ↾ 𝐵) ∈ V |
14 | 11, 11 | mpoex 7847 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))) ∈ V |
15 | 13, 14 | op2ndd 7769 | . . 3 ⊢ (𝑃 = 〈(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))〉 → (2nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))) |
16 | 7, 15 | syl 17 | . 2 ⊢ (𝜑 → (2nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))) |
17 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
18 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
19 | 17, 18 | oveq12d 7228 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆)) |
20 | 19 | reseq2d 5848 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (1st ↾ (𝑥𝐻𝑦)) = (1st ↾ (𝑅𝐻𝑆))) |
21 | 1stf1.p | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝐵) | |
22 | 1stf2.p | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝐵) | |
23 | ovex 7243 | . . . 4 ⊢ (𝑅𝐻𝑆) ∈ V | |
24 | resfunexg 7028 | . . . 4 ⊢ ((Fun 1st ∧ (𝑅𝐻𝑆) ∈ V) → (1st ↾ (𝑅𝐻𝑆)) ∈ V) | |
25 | 10, 23, 24 | mp2an 692 | . . 3 ⊢ (1st ↾ (𝑅𝐻𝑆)) ∈ V |
26 | 25 | a1i 11 | . 2 ⊢ (𝜑 → (1st ↾ (𝑅𝐻𝑆)) ∈ V) |
27 | 16, 20, 21, 22, 26 | ovmpod 7358 | 1 ⊢ (𝜑 → (𝑅(2nd ‘𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3405 〈cop 4544 ↾ cres 5550 Fun wfun 6371 –onto→wfo 6375 ‘cfv 6377 (class class class)co 7210 ∈ cmpo 7212 1st c1st 7756 2nd c2nd 7757 Basecbs 16757 Hom chom 16810 Catccat 17164 ×c cxpc 17672 1stF c1stf 17673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-slot 16732 df-ndx 16742 df-base 16758 df-hom 16823 df-cco 16824 df-xpc 17676 df-1stf 17677 |
This theorem is referenced by: 1stfcl 17701 prf1st 17708 1st2ndprf 17710 uncf2 17742 diag12 17749 diag2 17750 yonedalem22 17783 |
Copyright terms: Public domain | W3C validator |