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Mirrors > Home > MPE Home > Th. List > trcleq2lem | Structured version Visualization version GIF version |
Description: Equality implies bijection. (Contributed by RP, 5-May-2020.) |
Ref | Expression |
---|---|
trcleq2lem | ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3903 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵)) | |
2 | id 22 | . . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
3 | 2, 2 | coeq12d 5707 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵)) |
4 | 3, 2 | sseq12d 3910 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∘ 𝐴) ⊆ 𝐴 ↔ (𝐵 ∘ 𝐵) ⊆ 𝐵)) |
5 | 1, 4 | anbi12d 634 | 1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ⊆ wss 3843 ∘ ccom 5529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-in 3850 df-ss 3860 df-br 5031 df-opab 5093 df-co 5534 |
This theorem is referenced by: cvbtrcl 14443 trcleq12lem 14444 trclublem 14446 cotrtrclfv 14463 trclun 14465 trclexi 40795 dftrcl3 40896 |
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