Theorem invrcl2 49654

Description: Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.)

Hypotheses

Ref	Expression
invrcl.n	⊢ 𝑁 = (Inv‘𝐶)
invrcl.f	⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)
invrcl2.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
invrcl2	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Proof of Theorem invrcl2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		invrcl.f	. . . 4 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)
2		df-br 5106	. . . 4 ⊢ (𝐹(𝑋𝑁𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌))
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌))
4		invrcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		invrcl.n	. . . . 5 ⊢ 𝑁 = (Inv‘𝐶)
6	5, 1	invrcl 49653	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2765	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8	4, 5, 6, 7	invffval 17805	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
9	8	oveqd 7417	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
10	3, 9	eleqtrd 2867	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
11		eqid 2765	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))
12	11	elmpocl 7641	. 2 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
13	10, 12	syl 18	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∩ cin 3906  ⟨cop 4591   class class class wbr 5105  ^◡ccnv 5651  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17259  Sectcsect 17791  Invcinv 17792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-inv 17795

This theorem is referenced by:  isinv2  49655  isoval2  49664