Theorem negslem1 42172

Description: An equivalence between identically restricted order-reversing self-isometries. (Contributed by RP, 30-Sep-2024.)

Assertion

Ref	Expression
negslem1	⊢ (𝐴 = 𝐵 → ((𝐹 ↾ 𝐴) Isom 𝑅, ◡𝑅(𝐴, 𝐴) ↔ (𝐹 ↾ 𝐵) Isom 𝑅, ◡𝑅(𝐵, 𝐵)))

Proof of Theorem negslem1

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2	1, 1	resisoeq45d 42171	1 ⊢ (𝐴 = 𝐵 → ((𝐹 ↾ 𝐴) Isom 𝑅, ◡𝑅(𝐴, 𝐴) ↔ (𝐹 ↾ 𝐵) Isom 𝑅, ◡𝑅(𝐵, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542  ^◡ccnv 5676   ↾ cres 5679   Isom wiso 6545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553

This theorem is referenced by: (None)