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Mirrors > Home > MPE Home > Th. List > fodomnum | Structured version Visualization version GIF version |
Description: A version of fodom 10548 that doesn't require the Axiom of Choice ax-ac 10484. (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 7960 | . . . . 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 10074 | . . . 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 10081 | . 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 2098 Vcvv 3461 class class class wbr 5149 dom cdm 5678 –onto→wfo 6547 ≼ cdom 8962 cardccrd 9960 AC wacn 9963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-card 9964 df-acn 9967 |
This theorem is referenced by: fonum 10083 fodomfi2 10085 infpwfien 10087 inffien 10088 wdomnumr 10089 iunfictbso 10139 infmap2 10243 fictb 10270 cfflb 10284 cfslb2n 10293 fodomg 10547 rankcf 10802 tskuni 10808 tskurn 10814 znnen 16192 qnnen 16193 cygctb 19859 1stcrestlem 23400 2ndcctbss 23403 2ndcomap 23406 2ndcsep 23407 tx1stc 23598 tx2ndc 23599 met1stc 24474 met2ndci 24475 re2ndc 24761 uniiccdif 25551 dyadmbl 25573 opnmblALT 25576 mbfimaopnlem 25628 aannenlem3 26310 rn1st 44785 |
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