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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 6249 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelexi 4402 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
3 | 2 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 → 𝐴 ∈ V)) |
4 | f1f 5112 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
5 | fdm 5070 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
6 | vex 2604 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
7 | 6 | dmex 4616 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
8 | 5, 7 | syl6eqelr 2170 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
9 | 4, 8 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
10 | 9 | exlimiv 1529 | . . 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 5108 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦)) | |
13 | 12 | exbidv 1746 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)) |
14 | f1eq3 5109 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵)) | |
15 | 14 | exbidv 1746 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
16 | df-dom 6246 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
17 | 13, 15, 16 | brabg 4024 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
18 | 17 | expcom 114 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
19 | 3, 11, 18 | pm5.21ndd 653 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 class class class wbr 3785 dom cdm 4363 ⟶wf 4918 –1-1→wf1 4919 ≼ cdom 6243 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 df-fn 4925 df-f 4926 df-f1 4927 df-dom 6246 |
This theorem is referenced by: brdomi 6253 brdom 6254 f1dom2g 6259 f1domg 6261 dom3d 6277 phplem4dom 6348 |
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