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Theorem invfval 16413
Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
invfval.s 𝑆 = (Sect‘𝐶)
Assertion
Ref Expression
invfval (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))

Proof of Theorem invfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.b . . 3 𝐵 = (Base‘𝐶)
2 invfval.n . . 3 𝑁 = (Inv‘𝐶)
3 invfval.c . . 3 (𝜑𝐶 ∈ Cat)
4 invfval.x . . 3 (𝜑𝑋𝐵)
5 invfval.s . . 3 𝑆 = (Sect‘𝐶)
61, 2, 3, 4, 4, 5invffval 16412 . 2 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
7 simprl 794 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 796 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 6665 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌))
108, 7oveq12d 6665 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
1110cnveqd 5296 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
129, 11ineq12d 3813 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
13 invfval.y . 2 (𝜑𝑌𝐵)
14 ovex 6675 . . . 4 (𝑋𝑆𝑌) ∈ V
1514inex1 4797 . . 3 ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V
1615a1i 11 . 2 (𝜑 → ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V)
176, 12, 4, 13, 16ovmpt2d 6785 1 (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  cin 3571  ccnv 5111  cfv 5886  (class class class)co 6647  Basecbs 15851  Catccat 16319  Sectcsect 16398  Invcinv 16399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-inv 16402
This theorem is referenced by:  isinv  16414  invss  16415  dfiso2  16426  oppcinv  16434
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