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Mirrors > Home > MPE Home > Th. List > fnct | Structured version Visualization version GIF version |
Description: If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
fnct | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctex 8507 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
2 | 1 | adantl 485 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐴 ∈ V) |
3 | fndm 6425 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | 3 | eleq1d 2874 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)) |
5 | 4 | adantr 484 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)) |
6 | 2, 5 | mpbird 260 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → dom 𝐹 ∈ V) |
7 | fnfun 6423 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
8 | 7 | adantr 484 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → Fun 𝐹) |
9 | funrnex 7637 | . . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V)) | |
10 | 6, 8, 9 | sylc 65 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ∈ V) |
11 | 2, 10 | xpexd 7454 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V) |
12 | simpl 486 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 Fn 𝐴) | |
13 | dffn3 6499 | . . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) | |
14 | 12, 13 | sylib 221 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹) |
15 | fssxp 6508 | . . . 4 ⊢ (𝐹:𝐴⟶ran 𝐹 → 𝐹 ⊆ (𝐴 × ran 𝐹)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹)) |
17 | ssdomg 8538 | . . 3 ⊢ ((𝐴 × ran 𝐹) ∈ V → (𝐹 ⊆ (𝐴 × ran 𝐹) → 𝐹 ≼ (𝐴 × ran 𝐹))) | |
18 | 11, 16, 17 | sylc 65 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ (𝐴 × ran 𝐹)) |
19 | xpdom1g 8597 | . . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹)) | |
20 | 10, 19 | sylancom 591 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹)) |
21 | omex 9090 | . . . . 5 ⊢ ω ∈ V | |
22 | fnrndomg 9947 | . . . . . . 7 ⊢ (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴)) | |
23 | 2, 12, 22 | sylc 65 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ 𝐴) |
24 | domtr 8545 | . . . . . 6 ⊢ ((ran 𝐹 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω) | |
25 | 23, 24 | sylancom 591 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → ran 𝐹 ≼ ω) |
26 | xpdom2g 8596 | . . . . 5 ⊢ ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω)) | |
27 | 21, 25, 26 | sylancr 590 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω)) |
28 | domtr 8545 | . . . 4 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω)) | |
29 | 20, 27, 28 | syl2anc 587 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω)) |
30 | xpomen 9426 | . . 3 ⊢ (ω × ω) ≈ ω | |
31 | domentr 8551 | . . 3 ⊢ (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω) | |
32 | 29, 30, 31 | sylancl 589 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω) |
33 | domtr 8545 | . 2 ⊢ ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω) | |
34 | 18, 32, 33 | syl2anc 587 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 × cxp 5517 dom cdm 5519 ran crn 5520 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 ωcom 7560 ≈ cen 8489 ≼ cdom 8490 |
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: mptct 9949 mpocti 30477 mptctf 30479 omssubadd 31668 |
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