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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 17431 | . . 3 ⊢ (𝜑 → 𝑃 = 〈(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))〉) |
8 | fo1st 7700 | . . . . . 6 ⊢ 1st :V–onto→V | |
9 | fofun 6585 | . . . . . 6 ⊢ (1st :V–onto→V → Fun 1st ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ Fun 1st |
11 | 2 | fvexi 6678 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | resfunexg 6970 | . . . . 5 ⊢ ((Fun 1st ∧ 𝐵 ∈ V) → (1st ↾ 𝐵) ∈ V) | |
13 | 10, 11, 12 | mp2an 688 | . . . 4 ⊢ (1st ↾ 𝐵) ∈ V |
14 | 11, 11 | mpoex 7768 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))) ∈ V |
15 | 13, 14 | op2ndd 7691 | . . 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 767 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅) | |
18 | simprr 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆) | |
19 | 17, 18 | oveq12d 7163 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆)) |
20 | 19 | reseq2d 5847 | . 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 7178 | . . . 4 ⊢ (𝑅𝐻𝑆) ∈ V | |
24 | resfunexg 6970 | . . . 4 ⊢ ((Fun 1st ∧ (𝑅𝐻𝑆) ∈ V) → (1st ↾ (𝑅𝐻𝑆)) ∈ V) | |
25 | 10, 23, 24 | mp2an 688 | . . 3 ⊢ (1st ↾ (𝑅𝐻𝑆)) ∈ V |
26 | 25 | a1i 11 | . 2 ⊢ (𝜑 → (1st ↾ (𝑅𝐻𝑆)) ∈ V) |
27 | 16, 20, 21, 22, 26 | ovmpod 7291 | 1 ⊢ (𝜑 → (𝑅(2nd ‘𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3495 〈cop 4565 ↾ cres 5551 Fun wfun 6343 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7145 ∈ cmpo 7147 1st c1st 7678 2nd c2nd 7679 Basecbs 16473 Hom chom 16566 Catccat 16925 ×c cxpc 17408 1stF c1stf 17409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-ndx 16476 df-slot 16477 df-base 16479 df-hom 16579 df-cco 16580 df-xpc 17412 df-1stf 17413 |
This theorem is referenced by: 1stfcl 17437 prf1st 17444 1st2ndprf 17446 uncf2 17477 diag12 17484 diag2 17485 yonedalem22 17518 |
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