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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 18236 | . . 3 ⊢ (𝜑 → 𝑃 = 〈(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))〉) |
| 8 | fo1st 8034 | . . . . . 6 ⊢ 1st :V–onto→V | |
| 9 | fofun 6821 | . . . . . 6 ⊢ (1st :V–onto→V → Fun 1st ) | |
| 10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ Fun 1st |
| 11 | 2 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 12 | resfunexg 7235 | . . . . 5 ⊢ ((Fun 1st ∧ 𝐵 ∈ V) → (1st ↾ 𝐵) ∈ V) | |
| 13 | 10, 11, 12 | mp2an 692 | . . . 4 ⊢ (1st ↾ 𝐵) ∈ V |
| 14 | 11, 11 | mpoex 8104 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))) ∈ V |
| 15 | 13, 14 | op2ndd 8025 | . . 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 7449 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆)) |
| 20 | 19 | reseq2d 5997 | . 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 7464 | . . . 4 ⊢ (𝑅𝐻𝑆) ∈ V | |
| 24 | resfunexg 7235 | . . . 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 7585 | 1 ⊢ (𝜑 → (𝑅(2nd ‘𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↾ cres 5687 Fun wfun 6555 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 Basecbs 17247 Hom chom 17308 Catccat 17707 ×c cxpc 18213 1stF c1stf 18214 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-slot 17219 df-ndx 17231 df-base 17248 df-hom 17321 df-cco 17322 df-xpc 18217 df-1stf 18218 |
| This theorem is referenced by: 1stfcl 18242 prf1st 18249 1st2ndprf 18251 uncf2 18282 diag12 18289 diag2 18290 yonedalem22 18323 |
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