Theorem invrcl2 49512

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 5087	. . . 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 49511	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2737	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8	4, 5, 6, 7	invffval 17716	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
9	8	oveqd 7377	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
10	3, 9	eleqtrd 2839	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
11		eqid 2737	. . 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 7601	. 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 1542   ∈ wcel 2114   ∩ cin 3889  ⟨cop 4574   class class class wbr 5086  ^◡ccnv 5623  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  Basecbs 17170  Sectcsect 17702  Invcinv 17703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-inv 17706

This theorem is referenced by:  isinv2  49513  isoval2  49522