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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 8473 | . 2 ⊢ (𝐵 ≼* 𝐴 → (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) | |
2 | simpr 477 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 = ∅) | |
3 | 0fin 8188 | . . . . 5 ⊢ ∅ ∈ Fin | |
4 | finnum 8774 | . . . . 5 ⊢ (∅ ∈ Fin → ∅ ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ dom card |
6 | 2, 5 | syl6eqel 2709 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 ∈ dom card) |
7 | fonum 8881 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) | |
8 | 7 | ex 450 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
9 | 8 | exlimdv 1861 | . . . 4 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
10 | 9 | imp 445 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) |
11 | 6, 10 | jaodan 826 | . 2 ⊢ ((𝐴 ∈ dom card ∧ (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) → 𝐵 ∈ dom card) |
12 | 1, 11 | sylan2 491 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∅c0 3915 class class class wbr 4653 dom cdm 5114 –onto→wfo 5886 Fincfn 7955 ≼* cwdom 8462 cardccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-fin 7959 df-wdom 8464 df-card 8765 df-acn 8768 |
This theorem is referenced by: ptcmplem2 21857 |
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