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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 6517 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝑇) |
3 | unirnfdomd.3 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
4 | fnex 6982 | . . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V) | |
5 | 2, 3, 4 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | rnexg 7616 | . . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V) |
8 | frn 6522 | . . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin) | |
9 | dfss3 3958 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin) | |
10 | 8, 9 | sylib 220 | . . . . . 6 ⊢ (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin) |
11 | fict 9118 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω) | |
12 | 11 | ralimi 3162 | . . . . . 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 9967 | . . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ∪ ran 𝐹 ≼ (ran 𝐹 × ω)) | |
15 | 7, 13, 14 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (ran 𝐹 × ω)) |
16 | fnrndomg 9960 | . . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇)) | |
17 | 3, 2, 16 | sylc 65 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇) |
18 | omex 9108 | . . . . . 6 ⊢ ω ∈ V | |
19 | 18 | xpdom1 8618 | . . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω)) |
20 | 17, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω)) |
21 | domtr 8564 | . . . 4 ⊢ ((∪ ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ∪ ran 𝐹 ≼ (𝑇 × ω)) | |
22 | 15, 20, 21 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × ω)) |
23 | unirnfdomd.2 | . . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) | |
24 | infinf 9990 | . . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇)) | |
25 | 3, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇)) |
26 | 23, 25 | mpbid 234 | . . . 4 ⊢ (𝜑 → ω ≼ 𝑇) |
27 | xpdom2g 8615 | . . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇)) | |
28 | 3, 26, 27 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇)) |
29 | domtr 8564 | . . 3 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ∪ ran 𝐹 ≼ (𝑇 × 𝑇)) | |
30 | 22, 28, 29 | syl2anc 586 | . 2 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × 𝑇)) |
31 | infxpidm 9986 | . . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇) | |
32 | 26, 31 | syl 17 | . 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇) |
33 | domentr 8570 | . 2 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ∪ ran 𝐹 ≼ 𝑇) | |
34 | 30, 32, 33 | syl2anc 586 | 1 ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 × cxp 5555 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 Fincfn 8511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-card 9370 df-acn 9373 df-ac 9544 |
This theorem is referenced by: acsdomd 17793 |
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