Theorem resisoeq45d 42914

Description: Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)

Hypotheses

Ref	Expression
resisoeq45.4	⊢ (𝜑 → 𝐴 = 𝐶)
resisoeq45.5	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
resisoeq45d	⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Proof of Theorem resisoeq45d

Step	Hyp	Ref	Expression
1		resisoeq45.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
2	1	reseq2d 5979	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐹 ↾ 𝐶))
3		resisoeq45.5	. 2 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 1, 3	isoeq145d 42913	1 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ↾ cres 5674   Isom wiso 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551

This theorem is referenced by:  negslem1  42915