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Mirrors > Home > ILE Home > Th. List > ennnfonelemdc | GIF version |
Description: Lemma for ennnfone 12380. A direct consequence of fidcenumlemrk 6931. (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 4593 | . . . . 5 ⊢ ω ∈ On | |
5 | 4 | onelssi 4414 | . . . 4 ⊢ (𝑃 ∈ ω → 𝑃 ⊆ ω) |
6 | 3, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝑃 ⊆ ω) |
7 | fof 5420 | . . . . 5 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴) | |
8 | 2, 7 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐹:ω⟶𝐴) |
9 | 8, 3 | ffvelrnd 5632 | . . 3 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
10 | 1, 2, 3, 6, 9 | fidcenumlemrk 6931 | . 2 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ (𝐹 “ 𝑃) ∨ ¬ (𝐹‘𝑃) ∈ (𝐹 “ 𝑃))) |
11 | df-dc 830 | . 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 703 DECID wdc 829 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 ωcom 4574 “ cima 4614 ⟶wf 5194 –onto→wfo 5196 ‘cfv 5198 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 |
This theorem is referenced by: ennnfonelemg 12358 ennnfonelemp1 12361 ennnfonelemss 12365 ennnfonelemkh 12367 ennnfonelemhf1o 12368 |
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