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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 17560 | . . 3 ⊢ (𝜑 → 𝑄 = 〈(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))〉) |
8 | fo2nd 7735 | . . . . . 6 ⊢ 2nd :V–onto→V | |
9 | fofun 6593 | . . . . . 6 ⊢ (2nd :V–onto→V → Fun 2nd ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ Fun 2nd |
11 | 2 | fvexi 6688 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | resfunexg 6988 | . . . . 5 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V) | |
13 | 10, 11, 12 | mp2an 692 | . . . 4 ⊢ (2nd ↾ 𝐵) ∈ V |
14 | 11, 11 | mpoex 7803 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V |
15 | 13, 14 | op2ndd 7725 | . . 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 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
18 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
19 | 17, 18 | oveq12d 7188 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆)) |
20 | 19 | reseq2d 5825 | . 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 7203 | . . . 4 ⊢ (𝑅𝐻𝑆) ∈ V | |
24 | resfunexg 6988 | . . . 4 ⊢ ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V) | |
25 | 10, 23, 24 | mp2an 692 | . . 3 ⊢ (2nd ↾ (𝑅𝐻𝑆)) ∈ V |
26 | 25 | a1i 11 | . 2 ⊢ (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V) |
27 | 16, 20, 21, 22, 26 | ovmpod 7317 | 1 ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 〈cop 4522 ↾ cres 5527 Fun wfun 6333 –onto→wfo 6337 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 2nd c2nd 7713 Basecbs 16586 Hom chom 16679 Catccat 17038 ×c cxpc 17534 2ndF c2ndf 17536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-ndx 16589 df-slot 16590 df-base 16592 df-hom 16692 df-cco 16693 df-xpc 17538 df-2ndf 17540 |
This theorem is referenced by: 2ndfcl 17564 prf2nd 17571 1st2ndprf 17572 uncf2 17603 curf2ndf 17613 yonedalem22 17644 |
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