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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 8502 | . 2 ⊢ Rel ≈ | |
2 | f1of1 6607 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦) | |
3 | 2 | eximi 1826 | . . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦) |
4 | opabidw 5403 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
5 | opabidw 5403 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1→𝑦) | |
6 | 3, 4, 5 | 3imtr4i 293 | . . 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 8498 | . . . 4 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
8 | 7 | eleq2i 2901 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}) |
9 | df-dom 8499 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
10 | 9 | eleq2i 2901 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≼ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}) |
11 | 6, 8, 10 | 3imtr4i 293 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ → 〈𝑥, 𝑦〉 ∈ ≼ ) |
12 | 1, 11 | relssi 5653 | 1 ⊢ ≈ ⊆ ≼ |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1771 ∈ wcel 2105 ⊆ wss 3933 〈cop 4563 {copab 5119 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ≈ cen 8494 ≼ cdom 8495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-rel 5555 df-f1o 6355 df-en 8498 df-dom 8499 |
This theorem is referenced by: dfdom2 8523 endom 8524 |
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