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Mirrors > Home > ILE Home > Th. List > fidcenumlemim | GIF version |
Description: Lemma for fidcenum 7001. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.) |
Ref | Expression |
---|---|
fidcenumlemim | ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidceq 6912 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → DECID 𝑥 = 𝑦) | |
2 | 1 | 3expb 1206 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → DECID 𝑥 = 𝑦) |
3 | 2 | ralrimivva 2572 | . 2 ⊢ (𝐴 ∈ Fin → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
4 | isfi 6802 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) | |
5 | ensym 6822 | . . . . 5 ⊢ (𝐴 ≈ 𝑛 → 𝑛 ≈ 𝐴) | |
6 | bren 6788 | . . . . . 6 ⊢ (𝑛 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑛–1-1-onto→𝐴) | |
7 | f1ofo 5495 | . . . . . . 7 ⊢ (𝑓:𝑛–1-1-onto→𝐴 → 𝑓:𝑛–onto→𝐴) | |
8 | 7 | eximi 1611 | . . . . . 6 ⊢ (∃𝑓 𝑓:𝑛–1-1-onto→𝐴 → ∃𝑓 𝑓:𝑛–onto→𝐴) |
9 | 6, 8 | sylbi 121 | . . . . 5 ⊢ (𝑛 ≈ 𝐴 → ∃𝑓 𝑓:𝑛–onto→𝐴) |
10 | 5, 9 | syl 14 | . . . 4 ⊢ (𝐴 ≈ 𝑛 → ∃𝑓 𝑓:𝑛–onto→𝐴) |
11 | 10 | reximi 2587 | . . 3 ⊢ (∃𝑛 ∈ ω 𝐴 ≈ 𝑛 → ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴) |
12 | 4, 11 | sylbi 121 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴) |
13 | 3, 12 | jca 306 | 1 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 DECID wdc 835 ∃wex 1503 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 class class class wbr 4025 ωcom 4614 –onto→wfo 5240 –1-1-onto→wf1o 5241 ≈ cen 6779 Fincfn 6781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-tr 4124 df-id 4318 df-iord 4391 df-on 4393 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-er 6574 df-en 6782 df-fin 6784 |
This theorem is referenced by: fidcenum 7001 |
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