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Mirrors > Home > MPE Home > Th. List > Mathboxes > heeq12 | Structured version Visualization version GIF version |
Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
Ref | Expression |
---|---|
heeq12 | ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝑅 = 𝑆) | |
2 | simpr 487 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
3 | 1, 2 | imaeq12d 5929 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 “ 𝐴) = (𝑆 “ 𝐵)) |
4 | 3, 2 | sseq12d 3999 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → ((𝑅 “ 𝐴) ⊆ 𝐴 ↔ (𝑆 “ 𝐵) ⊆ 𝐵)) |
5 | df-he 40117 | . 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | |
6 | df-he 40117 | . 2 ⊢ (𝑆 hereditary 𝐵 ↔ (𝑆 “ 𝐵) ⊆ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 316 | 1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ⊆ wss 3935 “ cima 5557 hereditary whe 40116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-he 40117 |
This theorem is referenced by: heeq1 40121 heeq2 40122 frege77 40284 |
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