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Mirrors > Home > MPE Home > Th. List > unirnfdomd | Structured version Visualization version GIF version |
Description: The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
unirnfdomd.1 | ⊢ (𝜑 → 𝐹:𝑇⟶Fin) |
unirnfdomd.2 | ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) |
unirnfdomd.3 | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
unirnfdomd | ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unirnfdomd.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶Fin) | |
2 | 1 | ffnd 6488 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝑇) |
3 | unirnfdomd.3 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
4 | fnex 6957 | . . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V) | |
5 | 2, 3, 4 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | rnexg 7595 | . . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V) |
8 | frn 6493 | . . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin) | |
9 | dfss3 3903 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin) | |
10 | 8, 9 | sylib 221 | . . . . . 6 ⊢ (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin) |
11 | fict 9100 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω) | |
12 | 11 | ralimi 3128 | . . . . . 6 ⊢ (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) |
13 | 1, 10, 12 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) |
14 | unidom 9954 | . . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ∪ ran 𝐹 ≼ (ran 𝐹 × ω)) | |
15 | 7, 13, 14 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (ran 𝐹 × ω)) |
16 | fnrndomg 9947 | . . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇)) | |
17 | 3, 2, 16 | sylc 65 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇) |
18 | omex 9090 | . . . . . 6 ⊢ ω ∈ V | |
19 | 18 | xpdom1 8599 | . . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω)) |
20 | 17, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω)) |
21 | domtr 8545 | . . . 4 ⊢ ((∪ ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ∪ ran 𝐹 ≼ (𝑇 × ω)) | |
22 | 15, 20, 21 | syl2anc 587 | . . 3 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × ω)) |
23 | unirnfdomd.2 | . . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) | |
24 | infinf 9977 | . . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇)) | |
25 | 3, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇)) |
26 | 23, 25 | mpbid 235 | . . . 4 ⊢ (𝜑 → ω ≼ 𝑇) |
27 | xpdom2g 8596 | . . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇)) | |
28 | 3, 26, 27 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇)) |
29 | domtr 8545 | . . 3 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ∪ ran 𝐹 ≼ (𝑇 × 𝑇)) | |
30 | 22, 28, 29 | syl2anc 587 | . 2 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × 𝑇)) |
31 | infxpidm 9973 | . . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇) | |
32 | 26, 31 | syl 17 | . 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇) |
33 | domentr 8551 | . 2 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ∪ ran 𝐹 ≼ 𝑇) | |
34 | 30, 32, 33 | syl2anc 587 | 1 ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 × cxp 5517 ran crn 5520 Fn wfn 6319 ⟶wf 6320 ωcom 7560 ≈ cen 8489 ≼ cdom 8490 Fincfn 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-ac2 9874 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-card 9352 df-acn 9355 df-ac 9527 |
This theorem is referenced by: acsdomd 17783 |
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