![]() |
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 6542 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 4510 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
3 | 2 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 → 𝐴 ∈ V)) |
4 | f1f 5251 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
5 | fdm 5201 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
6 | vex 2636 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
7 | 6 | dmex 4731 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
8 | 5, 7 | syl6eqelr 2186 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
9 | 4, 8 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
10 | 9 | exlimiv 1541 | . . 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 5247 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦)) | |
13 | 12 | exbidv 1760 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)) |
14 | f1eq3 5248 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵)) | |
15 | 14 | exbidv 1760 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
16 | df-dom 6539 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
17 | 13, 15, 16 | brabg 4120 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
18 | 17 | expcom 115 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
19 | 3, 11, 18 | pm5.21ndd 659 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1296 ∃wex 1433 ∈ wcel 1445 Vcvv 2633 class class class wbr 3867 dom cdm 4467 ⟶wf 5045 –1-1→wf1 5046 ≼ cdom 6536 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-xp 4473 df-rel 4474 df-cnv 4475 df-dm 4477 df-rn 4478 df-fn 5052 df-f 5053 df-f1 5054 df-dom 6539 |
This theorem is referenced by: brdomi 6546 brdom 6547 f1dom2g 6553 f1domg 6555 dom3d 6571 phplem4dom 6658 djudom 6863 |
Copyright terms: Public domain | W3C validator |