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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 6516 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 4494 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
3 | 2 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 → 𝐴 ∈ V)) |
4 | f1f 5229 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
5 | fdm 5179 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
6 | vex 2623 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
7 | 6 | dmex 4712 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
8 | 5, 7 | syl6eqelr 2180 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
9 | 4, 8 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
10 | 9 | exlimiv 1535 | . . 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 5225 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦)) | |
13 | 12 | exbidv 1754 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)) |
14 | f1eq3 5226 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵)) | |
15 | 14 | exbidv 1754 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
16 | df-dom 6513 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
17 | 13, 15, 16 | brabg 4105 | . . 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 657 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1290 ∃wex 1427 ∈ wcel 1439 Vcvv 2620 class class class wbr 3851 dom cdm 4452 ⟶wf 5024 –1-1→wf1 5025 ≼ cdom 6510 |
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 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-xp 4458 df-rel 4459 df-cnv 4460 df-dm 4462 df-rn 4463 df-fn 5031 df-f 5032 df-f1 5033 df-dom 6513 |
This theorem is referenced by: brdomi 6520 brdom 6521 f1dom2g 6527 f1domg 6529 dom3d 6545 phplem4dom 6632 djudom 6837 |
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