1stf2 - Metamath Proof Explorer

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Theorem 1stf2 18393

Description: Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
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 1^st_F 𝐷)
1stf1.p	⊢ (𝜑 → 𝑅 ∈ 𝐵)
1stf2.p	⊢ (𝜑 → 𝑆 ∈ 𝐵)

**Assertion**
Ref	Expression
1stf2	⊢ (𝜑 → (𝑅(2^nd ‘𝑃)𝑆) = (1^st ↾ (𝑅𝐻𝑆)))

Proof of Theorem 1stf2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

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 1^st_F 𝐷)
7	1, 2, 3, 4, 5, 6	1stfval 18391	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩)
8		fo1st 8042	. . . . . 6 ⊢ 1^st :V⟶ontoV
9		fofun 6816	. . . . . 6 ⊢ (1^st :V⟶ontoV → Fun 1^st )
10	8, 9	ax-mp 5	. . . . 5 ⊢ Fun 1^st
11	2	fvexi 6919	. . . . 5 ⊢ 𝐵 ∈ V
12		resfunexg 7241	. . . . 5 ⊢ ((Fun 1^st ∧ 𝐵 ∈ V) → (1^st ↾ 𝐵) ∈ V)
13	10, 11, 12	mp2an 705	. . . 4 ⊢ (1^st ↾ 𝐵) ∈ V
14	11, 11	mpoex 8114	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))) ∈ V
15	13, 14	op2ndd 8033	. . 3 ⊢ (𝑃 = ⟨(1^st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦)))⟩ → (2^nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))))
16	7, 15	syl 18	. 2 ⊢ (𝜑 → (2^nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1^st ↾ (𝑥𝐻𝑦))))
17		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑥 = 𝑅)
18		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → 𝑦 = 𝑆)
19	17, 18	oveq12d 7458	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
20	19	reseq2d 5980	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑅 ∧ 𝑦 = 𝑆)) → (1^st ↾ (𝑥𝐻𝑦)) = (1^st ↾ (𝑅𝐻𝑆)))
21		1stf1.p	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
22		1stf2.p	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23		ovex 7473	. . . 4 ⊢ (𝑅𝐻𝑆) ∈ V
24		resfunexg 7241	. . . 4 ⊢ ((Fun 1^st ∧ (𝑅𝐻𝑆) ∈ V) → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
25	10, 23, 24	mp2an 705	. . 3 ⊢ (1^st ↾ (𝑅𝐻𝑆)) ∈ V
26	25	a1i 11	. 2 ⊢ (𝜑 → (1^st ↾ (𝑅𝐻𝑆)) ∈ V)
27	16, 20, 21, 22, 26	ovmpod 7593	1 ⊢ (𝜑 → (𝑅(2^nd ‘𝑃)𝑆) = (1^st ↾ (𝑅𝐻𝑆)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⟨cop 4838 ↾ cres 5657 Fun wfun 6549 ⟶ontowfo 6553 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 1^st c1st 8020 2^nd c2nd 8021 Basecbs 17413 Hom chom 17465 Catccat 17864 ×_c cxpc 18368 1^st_F c1stf 18369

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-er 8742 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-slot 17386 df-ndx 17398 df-base 17414 df-hom 17478 df-cco 17479 df-xpc 18372 df-1stf 18373

This theorem is used by: 1stfcl 18397 prf1st 18404 1st2ndprf 18406 uncf2 18437 diag12 18444 diag2 18445 yonedalem22 18478

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