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Mirrors > Home > MPE Home > Th. List > numwdom | Structured version Visualization version GIF version |
Description: A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
numwdom | ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴) → 𝐵 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brwdomi 9371 | . 2 ⊢ (𝐵 ≼* 𝐴 → (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) | |
2 | simpr 486 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 = ∅) | |
3 | 0fin 8992 | . . . . 5 ⊢ ∅ ∈ Fin | |
4 | finnum 9750 | . . . . 5 ⊢ (∅ ∈ Fin → ∅ ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ dom card |
6 | 2, 5 | eqeltrdi 2845 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 ∈ dom card) |
7 | fonum 9860 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) | |
8 | 7 | ex 414 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
9 | 8 | exlimdv 1934 | . . . 4 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
10 | 9 | imp 408 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) |
11 | 6, 10 | jaodan 956 | . 2 ⊢ ((𝐴 ∈ dom card ∧ (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) → 𝐵 ∈ dom card) |
12 | 1, 11 | sylan2 594 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 845 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ∅c0 4262 class class class wbr 5081 dom cdm 5600 –onto→wfo 6456 Fincfn 8764 ≼* cwdom 9367 cardccrd 9737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-fin 8768 df-wdom 9368 df-card 9741 df-acn 9744 |
This theorem is referenced by: ptcmplem2 23249 |
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