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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 6738 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝑇) |
3 | unirnfdomd.3 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
4 | fnex 7237 | . . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V) | |
5 | 2, 3, 4 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | rnexg 7925 | . . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V) |
8 | frn 6744 | . . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin) | |
9 | dfss3 3984 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin) | |
10 | 8, 9 | sylib 218 | . . . . . 6 ⊢ (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin) |
11 | fict 9691 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω) | |
12 | 11 | ralimi 3081 | . . . . . 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 10581 | . . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ∪ ran 𝐹 ≼ (ran 𝐹 × ω)) | |
15 | 7, 13, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (ran 𝐹 × ω)) |
16 | fnrndomg 10574 | . . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇)) | |
17 | 3, 2, 16 | sylc 65 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇) |
18 | omex 9681 | . . . . . 6 ⊢ ω ∈ V | |
19 | 18 | xpdom1 9110 | . . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω)) |
20 | 17, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω)) |
21 | domtr 9046 | . . . 4 ⊢ ((∪ ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ∪ ran 𝐹 ≼ (𝑇 × ω)) | |
22 | 15, 20, 21 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × ω)) |
23 | unirnfdomd.2 | . . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) | |
24 | infinf 10604 | . . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇)) | |
25 | 3, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇)) |
26 | 23, 25 | mpbid 232 | . . . 4 ⊢ (𝜑 → ω ≼ 𝑇) |
27 | xpdom2g 9107 | . . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇)) | |
28 | 3, 26, 27 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇)) |
29 | domtr 9046 | . . 3 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ∪ ran 𝐹 ≼ (𝑇 × 𝑇)) | |
30 | 22, 28, 29 | syl2anc 584 | . 2 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × 𝑇)) |
31 | infxpidm 10600 | . . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇) | |
32 | 26, 31 | syl 17 | . 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇) |
33 | domentr 9052 | . 2 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ∪ ran 𝐹 ≼ 𝑇) | |
34 | 30, 32, 33 | syl2anc 584 | 1 ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 × cxp 5687 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ωcom 7887 ≈ cen 8981 ≼ cdom 8982 Fincfn 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-card 9977 df-acn 9980 df-ac 10154 |
This theorem is referenced by: acsdomd 18615 |
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