Theorem issect 17762

Description: The property "𝐹 is a section of 𝐺". (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
issect.b	⊢ 𝐵 = (Base‘𝐶)
issect.h	⊢ 𝐻 = (Hom ‘𝐶)
issect.o	⊢ · = (comp‘𝐶)
issect.i	⊢ 1 = (Id‘𝐶)
issect.s	⊢ 𝑆 = (Sect‘𝐶)
issect.c	⊢ (𝜑 → 𝐶 ∈ Cat)
issect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
issect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
issect	⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋))))

Proof of Theorem issect

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issect.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		issect.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
3		issect.o	. . . 4 ⊢ · = (comp‘𝐶)
4		issect.i	. . . 4 ⊢ 1 = (Id‘𝐶)
5		issect.s	. . . 4 ⊢ 𝑆 = (Sect‘𝐶)
6		issect.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		issect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		issect.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 2, 3, 4, 5, 6, 7, 8	sectfval 17760	. . 3 ⊢ (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))})
10	9	breqd 5105	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺))
11		oveq12 7394	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
12	11	ancoms 461	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
13	12	eqeq1d 2758	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
14		eqid 2756	. . . 4 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))}
15	13, 14	brab2a 5733	. . 3 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
16		df-3an 1097	. . 3 ⊢ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
17	15, 16	bitr4i 280	. 2 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
18	10, 17	bitrdi 289	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ⟨cop 4582   class class class wbr 5094  {copab 5156  ‘cfv 6510  (class class class)co 7385  Basecbs 17221  Hom chom 17273  compcco 17274  Catccat 17672  Idccid 17673  Sectcsect 17753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-sect 17756

This theorem is referenced by:  issect2  17763  sectcan  17764  sectco  17765  oppcsect  17787  sectmon  17791  monsect  17792  funcsect  17881  fucsect  17984  invfuc  17986  setcsect  18098  catciso  18120  rngcsect  20658  ringcsect  20692  rngcsectALTV  48845  ringcsectALTV  48879  catcsect  49967  thincsect  50036