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Mirrors > Home > ILE Home > Th. List > enssdom | GIF version |
Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) |
Ref | Expression |
---|---|
enssdom | ⊢ ≈ ⊆ ≼ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 6225 | . 2 ⊢ Rel ≈ | |
2 | f1of1 5125 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦) | |
3 | 2 | eximi 1491 | . . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦) |
4 | opabid 3994 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
5 | opabid 3994 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1→𝑦) | |
6 | 3, 4, 5 | 3imtr4i 190 | . . 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 6222 | . . . 4 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
8 | 7 | eleq2i 2104 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}) |
9 | df-dom 6223 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
10 | 9 | eleq2i 2104 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ≼ ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}) |
11 | 6, 8, 10 | 3imtr4i 190 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ≈ → 〈𝑥, 𝑦〉 ∈ ≼ ) |
12 | 1, 11 | relssi 4431 | 1 ⊢ ≈ ⊆ ≼ |
Colors of variables: wff set class |
Syntax hints: ∃wex 1381 ∈ wcel 1393 ⊆ wss 2917 〈cop 3378 {copab 3817 –1-1→wf1 4899 –1-1-onto→wf1o 4901 ≈ cen 6219 ≼ cdom 6220 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 df-rel 4352 df-f1o 4909 df-en 6222 df-dom 6223 |
This theorem is referenced by: endom 6243 |
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