Theorem invrcl2 49266

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 5099	. . . 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 49265	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2736	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8	4, 5, 6, 7	invffval 17682	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
9	8	oveqd 7375	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
10	3, 9	eleqtrd 2838	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
11		eqid 2736	. . 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 7599	. 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 1541   ∈ wcel 2113   ∩ cin 3900  ⟨cop 4586   class class class wbr 5098  ^◡ccnv 5623  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17136  Sectcsect 17668  Invcinv 17669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-inv 17672

This theorem is referenced by:  isinv2  49267  isoval2  49276