![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fodomnum | Structured version Visualization version GIF version |
Description: A version of fodom 10460 that doesn't require the Axiom of Choice ax-ac 10396. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fodomnum | ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | focdmex 7889 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) | |
2 | 1 | com12 32 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ∈ dom card → 𝐵 ∈ V)) |
3 | numacn 9986 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 ∈ dom card → 𝐴 ∈ AC 𝐵)) | |
4 | 2, 3 | syli 39 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ∈ dom card → 𝐴 ∈ AC 𝐵)) |
5 | 4 | com12 32 | . 2 ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐴 ∈ AC 𝐵)) |
6 | fodomacn 9993 | . 2 ⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | |
7 | 5, 6 | syli 39 | 1 ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3446 class class class wbr 5106 dom cdm 5634 –onto→wfo 6495 ≼ cdom 8882 cardccrd 9872 AC wacn 9875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-card 9876 df-acn 9879 |
This theorem is referenced by: fonum 9995 fodomfi2 9997 infpwfien 9999 inffien 10000 wdomnumr 10001 iunfictbso 10051 infmap2 10155 fictb 10182 cfflb 10196 cfslb2n 10205 fodomg 10459 rankcf 10714 tskuni 10720 tskurn 10726 znnen 16095 qnnen 16096 cygctb 19670 1stcrestlem 22806 2ndcctbss 22809 2ndcomap 22812 2ndcsep 22813 tx1stc 23004 tx2ndc 23005 met1stc 23880 met2ndci 23881 re2ndc 24167 uniiccdif 24945 dyadmbl 24967 opnmblALT 24970 mbfimaopnlem 25022 aannenlem3 25693 rn1st 43509 |
Copyright terms: Public domain | W3C validator |