Theorem invrcl2 49011

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 5096	. . . 4 ⊢ (𝐹(𝑋𝑁𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌))
3	1, 2	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌))
4		invrcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		invrcl.n	. . . . 5 ⊢ 𝑁 = (Inv‘𝐶)
6	5, 1	invrcl 49010	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2729	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8	4, 5, 6, 7	invffval 17683	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
9	8	oveqd 7370	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
10	3, 9	eleqtrd 2830	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
11		eqid 2729	. . 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 7594	. 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 17	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3904  ⟨cop 4585   class class class wbr 5095  ^◡ccnv 5622  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  Basecbs 17138  Sectcsect 17669  Invcinv 17670

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-inv 17673

This theorem is referenced by:  isinv2  49012  isoval2  49021