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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 473 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝑅 = 𝑆) | |
2 | simpr 477 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
3 | 1, 2 | imaeq12d 5467 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 “ 𝐴) = (𝑆 “ 𝐵)) |
4 | 3, 2 | sseq12d 3634 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → ((𝑅 “ 𝐴) ⊆ 𝐴 ↔ (𝑆 “ 𝐵) ⊆ 𝐵)) |
5 | df-he 38067 | . 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | |
6 | df-he 38067 | . 2 ⊢ (𝑆 hereditary 𝐵 ↔ (𝑆 “ 𝐵) ⊆ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 303 | 1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ⊆ wss 3574 “ cima 5117 hereditary whe 38066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-he 38067 |
This theorem is referenced by: heeq1 38071 heeq2 38072 frege77 38234 |
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