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Mirrors > Home > MPE Home > Th. List > Mathboxes > ecxpid | Structured version Visualization version GIF version |
Description: The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
Ref | Expression |
---|---|
ecxpid | ⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3496 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | elecg 8329 | . . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥)) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥)) |
4 | brxp 5598 | . . . 4 ⊢ (𝑋(𝐴 × 𝐴)𝑥 ↔ (𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
5 | 4 | baib 538 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑋(𝐴 × 𝐴)𝑥 ↔ 𝑥 ∈ 𝐴)) |
6 | 3, 5 | bitrd 281 | . 2 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)) |
7 | 6 | eqrdv 2818 | 1 ⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 Vcvv 3493 class class class wbr 5063 × cxp 5550 [cec 8284 |
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 5200 ax-nul 5207 ax-pr 5327 |
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-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-br 5064 df-opab 5126 df-xp 5558 df-cnv 5560 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-ec 8288 |
This theorem is referenced by: qsxpid 30948 |
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