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Mirrors > Home > ILE Home > Th. List > domen | GIF version |
Description: Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
bren.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
domen | ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | brdom 6231 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | vex 2560 | . . . . . 6 ⊢ 𝑓 ∈ V | |
4 | 3 | f11o 5159 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
5 | 4 | exbii 1496 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
6 | excom 1554 | . . . 4 ⊢ (∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) | |
7 | 5, 6 | bitri 173 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
8 | bren 6228 | . . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥) | |
9 | 8 | anbi1i 431 | . . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
10 | 19.41v 1782 | . . . . 5 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) | |
11 | 9, 10 | bitr4i 176 | . . . 4 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
12 | 11 | exbii 1496 | . . 3 ⊢ (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
13 | 7, 12 | bitr4i 176 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
14 | 2, 13 | bitri 173 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 class class class wbr 3764 –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-13 1404 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 ax-un 4170 |
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-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-en 6222 df-dom 6223 |
This theorem is referenced by: domeng 6233 php5dom 6325 |
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