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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 8497 | . 2 ⊢ Rel ≈ | |
2 | f1of1 6589 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦) | |
3 | 2 | eximi 1836 | . . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦) |
4 | opabidw 5377 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
5 | opabidw 5377 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1→𝑦) | |
6 | 3, 4, 5 | 3imtr4i 295 | . . 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 8493 | . . . 4 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
8 | 7 | eleq2i 2881 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}) |
9 | df-dom 8494 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
10 | 9 | eleq2i 2881 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≼ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}) |
11 | 6, 8, 10 | 3imtr4i 295 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ → 〈𝑥, 𝑦〉 ∈ ≼ ) |
12 | 1, 11 | relssi 5624 | 1 ⊢ ≈ ⊆ ≼ |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1781 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 {copab 5092 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ≈ cen 8489 ≼ cdom 8490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-rel 5526 df-f1o 6331 df-en 8493 df-dom 8494 |
This theorem is referenced by: dfdom2 8518 endom 8519 |
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