Theorem heeq12 43800

Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
heeq12	⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐵))

Proof of Theorem heeq12

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝑅 = 𝑆)
2		simpr 484	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
3	1, 2	imaeq12d 6048	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 “ 𝐴) = (𝑆 “ 𝐵))
4	3, 2	sseq12d 3992	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → ((𝑅 “ 𝐴) ⊆ 𝐴 ↔ (𝑆 “ 𝐵) ⊆ 𝐵))
5		df-he 43797	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
6		df-he 43797	. 2 ⊢ (𝑆 hereditary 𝐵 ↔ (𝑆 “ 𝐵) ⊆ 𝐵)
7	4, 5, 6	3bitr4g 314	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ⊆ wss 3926   “ cima 5657   hereditary whe 43796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-he 43797

This theorem is referenced by:  heeq1  43801  heeq2  43802  frege77  43964