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Theorem invfval 17021
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 17020 . 2 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
7 simprl 769 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 771 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 7166 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌))
108, 7oveq12d 7166 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
1110cnveqd 5739 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
129, 11ineq12d 4188 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
13 invfval.y . 2 (𝜑𝑌𝐵)
14 ovex 7181 . . . 4 (𝑋𝑆𝑌) ∈ V
1514inex1 5212 . . 3 ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V
1615a1i 11 . 2 (𝜑 → ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V)
176, 12, 4, 13, 16ovmpod 7294 1 (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  Vcvv 3493  cin 3933  ccnv 5547  cfv 6348  (class class class)co 7148  Basecbs 16475  Catccat 16927  Sectcsect 17006  Invcinv 17007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-inv 17010
This theorem is referenced by:  isinv  17022  invss  17023  dfiso2  17034  oppcinv  17042
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