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Mirrors > Home > MPE Home > Th. List > Mathboxes > parteq1 | Structured version Visualization version GIF version |
Description: Equality theorem for partition. (Contributed by Peter Mazsa, 5-Oct-2021.) |
Ref | Expression |
---|---|
parteq1 | ⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjdmqseqeq1 37412 | . 2 ⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴))) | |
2 | dfpart2 37444 | . 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
3 | dfpart2 37444 | . 2 ⊢ (𝑆 Part 𝐴 ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)) | |
4 | 1, 2, 3 | 3bitr4g 313 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 dom cdm 5669 / cqs 8685 Disj wdisjALTV 36882 Part wpart 36887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8688 df-qs 8692 df-coss 37086 df-cnvrefrel 37202 df-dmqs 37314 df-funALTV 37357 df-disjALTV 37380 df-part 37441 |
This theorem is referenced by: parteq12 37451 parteq1i 37452 parteq1d 37453 |
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