Theorem invfval 17728

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	⊢ (𝜑 → 𝐶 ∈ Cat)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
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.

Step	Hyp	Ref	Expression
1		invfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invfval.s	. . 3 ⊢ 𝑆 = (Sect‘𝐶)
5	1, 2, 3, 4	invffval 17727	. 2 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
6		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
7		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
8	6, 7	oveq12d 7408	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌))
9	7, 6	oveq12d 7408	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
10	9	cnveqd 5842	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ◡(𝑦𝑆𝑥) = ◡(𝑌𝑆𝑋))
11	8, 10	ineq12d 4187	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)))
12		invfval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
13		invfval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		ovex 7423	. . . 4 ⊢ (𝑋𝑆𝑌) ∈ V
15	14	inex1 5275	. . 3 ⊢ ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V)
17	5, 11, 12, 13, 16	ovmpod 7544	1 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916  ^◡ccnv 5640  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Catccat 17632  Sectcsect 17713  Invcinv 17714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-inv 17717

This theorem is referenced by:  isinv  17729  invss  17730  dfiso2  17741  oppcinv  17749  invpropdlem  49031