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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 6574 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | vex 2644 | . . . . . 6 ⊢ 𝑓 ∈ V | |
4 | 3 | f11o 5334 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
5 | 4 | exbii 1552 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
6 | excom 1610 | . . . 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 183 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
8 | bren 6571 | . . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥) | |
9 | 8 | anbi1i 449 | . . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
10 | 19.41v 1841 | . . . . 5 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) | |
11 | 9, 10 | bitr4i 186 | . . . 4 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
12 | 11 | exbii 1552 | . . 3 ⊢ (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
13 | 7, 12 | bitr4i 186 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
14 | 2, 13 | bitri 183 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1436 ∈ wcel 1448 Vcvv 2641 ⊆ wss 3021 class class class wbr 3875 –1-1→wf1 5056 –1-1-onto→wf1o 5058 ≈ cen 6562 ≼ cdom 6563 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-xp 4483 df-rel 4484 df-cnv 4485 df-dm 4487 df-rn 4488 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-en 6565 df-dom 6566 |
This theorem is referenced by: domeng 6576 php5dom 6686 |
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