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Mirrors > Home > MPE Home > Th. List > enssdom | Structured version Visualization version GIF version |
Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) |
Ref | Expression |
---|---|
enssdom | ⊢ ≈ ⊆ ≼ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 8988 | . 2 ⊢ Rel ≈ | |
2 | f1of1 6847 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦) | |
3 | 2 | eximi 1831 | . . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦) |
4 | opabidw 5533 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
5 | opabidw 5533 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1→𝑦) | |
6 | 3, 4, 5 | 3imtr4i 292 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} → 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}) |
7 | df-en 8984 | . . . 4 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
8 | 7 | eleq2i 2830 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}) |
9 | df-dom 8985 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
10 | 9 | eleq2i 2830 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≼ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}) |
11 | 6, 8, 10 | 3imtr4i 292 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ → 〈𝑥, 𝑦〉 ∈ ≼ ) |
12 | 1, 11 | relssi 5799 | 1 ⊢ ≈ ⊆ ≼ |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1775 ∈ wcel 2105 ⊆ wss 3962 〈cop 4636 {copab 5209 –1-1→wf1 6559 –1-1-onto→wf1o 6561 ≈ cen 8980 ≼ cdom 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5210 df-xp 5694 df-rel 5695 df-f1o 6569 df-en 8984 df-dom 8985 |
This theorem is referenced by: dfdom2 9016 endom 9017 |
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