Theorem invrcl2 49610

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 5100	. . . 4 ⊢ (𝐹(𝑋𝑁𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌))
3	1, 2	sylib 220	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌))
4		invrcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		invrcl.n	. . . . 5 ⊢ 𝑁 = (Inv‘𝐶)
6	5, 1	invrcl 49609	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2761	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8	4, 5, 6, 7	invffval 17774	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
9	8	oveqd 7409	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
10	3, 9	eleqtrd 2863	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))𝑌))
11		eqid 2761	. . 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 7633	. 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 399   = wceq 1559   ∈ wcel 2141   ∩ cin 3903  ⟨cop 4587   class class class wbr 5099  ^◡ccnv 5644  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Basecbs 17228  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:  isinv2  49611  isoval2  49620