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Mirrors > Home > ILE Home > Th. List > domtr | GIF version |
Description: Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
domtr | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6632 | . 2 ⊢ Rel ≼ | |
2 | vex 2684 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 2 | brdom 6637 | . . 3 ⊢ (𝑥 ≼ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1→𝑦) |
4 | vex 2684 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 4 | brdom 6637 | . . 3 ⊢ (𝑦 ≼ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1→𝑧) |
6 | eeanv 1902 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) ↔ (∃𝑔 𝑔:𝑥–1-1→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1→𝑧)) | |
7 | f1co 5335 | . . . . . . . 8 ⊢ ((𝑓:𝑦–1-1→𝑧 ∧ 𝑔:𝑥–1-1→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1→𝑧) | |
8 | 7 | ancoms 266 | . . . . . . 7 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1→𝑧) |
9 | vex 2684 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
10 | vex 2684 | . . . . . . . . 9 ⊢ 𝑔 ∈ V | |
11 | 9, 10 | coex 5079 | . . . . . . . 8 ⊢ (𝑓 ∘ 𝑔) ∈ V |
12 | f1eq1 5318 | . . . . . . . 8 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ:𝑥–1-1→𝑧 ↔ (𝑓 ∘ 𝑔):𝑥–1-1→𝑧)) | |
13 | 11, 12 | spcev 2775 | . . . . . . 7 ⊢ ((𝑓 ∘ 𝑔):𝑥–1-1→𝑧 → ∃ℎ ℎ:𝑥–1-1→𝑧) |
14 | 8, 13 | syl 14 | . . . . . 6 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → ∃ℎ ℎ:𝑥–1-1→𝑧) |
15 | 4 | brdom 6637 | . . . . . 6 ⊢ (𝑥 ≼ 𝑧 ↔ ∃ℎ ℎ:𝑥–1-1→𝑧) |
16 | 14, 15 | sylibr 133 | . . . . 5 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
17 | 16 | exlimivv 1868 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
18 | 6, 17 | sylbir 134 | . . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
19 | 3, 5, 18 | syl2anb 289 | . 2 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑧) → 𝑥 ≼ 𝑧) |
20 | 1, 19 | vtoclr 4582 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1468 class class class wbr 3924 ∘ ccom 4538 –1-1→wf1 5115 ≼ cdom 6626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-dom 6629 |
This theorem is referenced by: endomtr 6677 domentr 6678 cnvct 6696 ssct 6705 nndomo 6751 infnfi 6782 xpct 11898 |
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