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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 8990 | . 2 ⊢ Rel ≈ | |
| 2 | f1of1 6847 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦) | |
| 3 | 2 | eximi 1835 | . . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦) |
| 4 | opabidw 5529 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
| 5 | opabidw 5529 | . . . 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 8986 | . . . 4 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
| 8 | 7 | eleq2i 2833 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}) |
| 9 | df-dom 8987 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
| 10 | 9 | eleq2i 2833 | . . 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 5797 | 1 ⊢ ≈ ⊆ ≼ |
| Colors of variables: wff setvar class |
| Syntax hints: ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 {copab 5205 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ≈ cen 8982 ≼ cdom 8983 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 df-f1o 6568 df-en 8986 df-dom 8987 |
| This theorem is referenced by: dfdom2 9018 endom 9019 |
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