Theorem invffval 17774

Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) Removed redundant hypotheses. (Revised by Zhi Wang, 27-Oct-2025.)

Hypotheses

Ref	Expression
invfval.b	⊢ 𝐵 = (Base‘𝐶)
invfval.n	⊢ 𝑁 = (Inv‘𝐶)
invfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
invffval.s	⊢ 𝑆 = (Sect‘𝐶)

Assertion

Ref	Expression
invffval	⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝜑,𝑥,𝑦   𝑥,𝐶,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem invffval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.n	. 2 ⊢ 𝑁 = (Inv‘𝐶)
2		invfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fveq2 6863	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		invfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2814	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fveq2 6863	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
7		invffval.s	. . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶)
8	6, 7	eqtr4di 2814	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆)
9	8	oveqd 7409	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦))
10	8	oveqd 7409	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
11	10	cnveqd 5845	. . . . . 6 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦𝑆𝑥))
12	9, 11	ineq12d 4173	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))
13	5, 5, 12	mpoeq123dv 7467	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
14		df-inv 17764	. . . 4 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
15	4	fvexi 6877	. . . . 5 ⊢ 𝐵 ∈ V
16	15, 15	mpoex 8056	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) ∈ V
17	13, 14, 16	fvmpt 6971	. . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
18	2, 17	syl 17	. 2 ⊢ (𝜑 → (Inv‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
19	1, 18	eqtrid 2808	1 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141   ∩ cin 3903  ^◡ccnv 5644  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Basecbs 17228  Catccat 17679  Sectcsect 17760  Invcinv 17761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-inv 17764

This theorem is referenced by:  invfval  17775  isoval  17781  invrcl2  49610  invpropdlem  49623