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Mirrors > Home > MPE Home > Th. List > tposconst | Structured version Visualization version GIF version |
Description: The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.) |
Ref | Expression |
---|---|
tposconst | ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstmpo 7262 | . . 3 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | tposmpo 7922 | . 2 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶) |
3 | fconstmpo 7262 | . 2 ⊢ ((𝐵 × 𝐴) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | 2, 3 | eqtr4i 2846 | 1 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 {csn 4560 × cxp 5546 ∈ cmpo 7151 tpos ctpos 7884 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-oprab 7153 df-mpo 7154 df-tpos 7885 |
This theorem is referenced by: mattposvs 21059 |
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