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Mirrors > Home > ILE Home > Th. List > ennnfonelemdc | GIF version |
Description: Lemma for ennnfone 11938. A direct consequence of fidcenumlemrk 6842. (Contributed by Jim Kingdon, 15-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemdc.dceq | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
ennnfonelemdc.f | ⊢ (𝜑 → 𝐹:ω–onto→𝐴) |
ennnfonelemdc.p | ⊢ (𝜑 → 𝑃 ∈ ω) |
Ref | Expression |
---|---|
ennnfonelemdc | ⊢ (𝜑 → DECID (𝐹‘𝑃) ∈ (𝐹 “ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemdc.dceq | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) | |
2 | ennnfonelemdc.f | . . 3 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
3 | ennnfonelemdc.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ ω) | |
4 | omelon 4522 | . . . . 5 ⊢ ω ∈ On | |
5 | 4 | onelssi 4351 | . . . 4 ⊢ (𝑃 ∈ ω → 𝑃 ⊆ ω) |
6 | 3, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝑃 ⊆ ω) |
7 | fof 5345 | . . . . 5 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴) | |
8 | 2, 7 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐹:ω⟶𝐴) |
9 | 8, 3 | ffvelrnd 5556 | . . 3 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
10 | 1, 2, 3, 6, 9 | fidcenumlemrk 6842 | . 2 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ (𝐹 “ 𝑃) ∨ ¬ (𝐹‘𝑃) ∈ (𝐹 “ 𝑃))) |
11 | df-dc 820 | . 2 ⊢ (DECID (𝐹‘𝑃) ∈ (𝐹 “ 𝑃) ↔ ((𝐹‘𝑃) ∈ (𝐹 “ 𝑃) ∨ ¬ (𝐹‘𝑃) ∈ (𝐹 “ 𝑃))) | |
12 | 10, 11 | sylibr 133 | 1 ⊢ (𝜑 → DECID (𝐹‘𝑃) ∈ (𝐹 “ 𝑃)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 697 DECID wdc 819 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 ωcom 4504 “ cima 4542 ⟶wf 5119 –onto→wfo 5121 ‘cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 |
This theorem is referenced by: ennnfonelemg 11916 ennnfonelemp1 11919 ennnfonelemss 11923 ennnfonelemkh 11925 ennnfonelemhf1o 11926 |
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