![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > brdomg | GIF version |
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
brdomg | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6759 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 4681 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
3 | 2 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 → 𝐴 ∈ V)) |
4 | f1f 5433 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
5 | fdm 5383 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
6 | vex 2752 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
7 | 6 | dmex 4905 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
8 | 5, 7 | eqeltrrdi 2279 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
9 | 4, 8 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
10 | 9 | exlimiv 1608 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
11 | 10 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝐶 → (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)) |
12 | f1eq2 5429 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦)) | |
13 | 12 | exbidv 1835 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)) |
14 | f1eq3 5430 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵)) | |
15 | 14 | exbidv 1835 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
16 | df-dom 6756 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
17 | 13, 15, 16 | brabg 4281 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
18 | 17 | expcom 116 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
19 | 3, 11, 18 | pm5.21ndd 706 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1363 ∃wex 1502 ∈ wcel 2158 Vcvv 2749 class class class wbr 4015 dom cdm 4638 ⟶wf 5224 –1-1→wf1 5225 ≼ cdom 6753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-rel 4645 df-cnv 4646 df-dm 4648 df-rn 4649 df-fn 5231 df-f 5232 df-f1 5233 df-dom 6756 |
This theorem is referenced by: brdomi 6763 brdom 6764 f1dom2g 6770 f1domg 6772 dom3d 6788 phplem4dom 6876 djudom 7106 difinfsn 7113 djudoml 7232 djudomr 7233 nninfdc 12468 |
Copyright terms: Public domain | W3C validator |