Theorem invfun 17700

Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
invfval.b	⊢ 𝐵 = (Base‘𝐶)
invfval.n	⊢ 𝑁 = (Inv‘𝐶)
invfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
invss.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invss.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
invfun	⊢ (𝜑 → Fun (𝑋𝑁𝑌))

Proof of Theorem invfun

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

Step	Hyp	Ref	Expression
1		invfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. . . 4 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invss.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invss.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		eqid 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7	1, 2, 3, 4, 5, 6	invss 17697	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)))
8		relxp 5650	. . 3 ⊢ Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))
9		relss 5739	. . 3 ⊢ ((𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → (Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → Rel (𝑋𝑁𝑌)))
10	7, 8, 9	mpisyl 21	. 2 ⊢ (𝜑 → Rel (𝑋𝑁𝑌))
11		eqid 2737	. . . . . 6 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
12	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝐶 ∈ Cat)
13	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑌 ∈ 𝐵)
14	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑋 ∈ 𝐵)
15	1, 2, 3, 4, 5, 11	isinv 17696	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔 ∧ 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)))
16	15	simplbda 499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
17	16	adantrr 718	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
18	1, 2, 3, 4, 5, 11	isinv 17696	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)ℎ ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)ℎ ∧ ℎ(𝑌(Sect‘𝐶)𝑋)𝑓)))
19	18	simprbda 498	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
20	19	adantrl 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
21	1, 11, 12, 13, 14, 17, 20	sectcan 17691	. . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔 = ℎ)
22	21	ex 412	. . . 4 ⊢ (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
23	22	alrimiv 1929	. . 3 ⊢ (𝜑 → ∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
24	23	alrimivv 1930	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
25		dffun2 6510	. 2 ⊢ (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)))
26	10, 24, 25	sylanbrc 584	1 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903   class class class wbr 5100   × cxp 5630  Rel wrel 5637  Fun wfun 6494  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  Catccat 17599  Sectcsect 17680  Invcinv 17681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-cat 17603  df-cid 17604  df-sect 17683  df-inv 17684

This theorem is referenced by:  inviso1  17702  invf  17704  invco  17707  idinv  17725  ffthiso  17867  fuciso  17914  setciso  18027  catciso  18047  rngciso  20583  ringciso  20617  rngcisoALTV  48637  ringcisoALTV  48671