| Step | Hyp | Ref
| Expression |
| 1 | | ramub1.3 |
. . 3
⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) |
| 2 | | ramub1.m |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 3 | | nnm1nn0 12567 |
. . . 4
⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈
ℕ0) |
| 4 | 2, 3 | syl 17 |
. . 3
⊢ (𝜑 → (𝑀 − 1) ∈
ℕ0) |
| 5 | | ramub1.r |
. . 3
⊢ (𝜑 → 𝑅 ∈ Fin) |
| 6 | | ramub1.1 |
. . 3
⊢ (𝜑 → 𝐺:𝑅⟶ℕ0) |
| 7 | | ramub1.2 |
. . 3
⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈
ℕ0) |
| 8 | | ramub1.4 |
. . . 4
⊢ (𝜑 → 𝑆 ∈ Fin) |
| 9 | | diffi 9215 |
. . . 4
⊢ (𝑆 ∈ Fin → (𝑆 ∖ {𝑋}) ∈ Fin) |
| 10 | 8, 9 | syl 17 |
. . 3
⊢ (𝜑 → (𝑆 ∖ {𝑋}) ∈ Fin) |
| 11 | 7 | nn0red 12588 |
. . . . 5
⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℝ) |
| 12 | 11 | leidd 11829 |
. . . 4
⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ ((𝑀 − 1) Ramsey 𝐺)) |
| 13 | | hashcl 14395 |
. . . . . . 7
⊢ ((𝑆 ∖ {𝑋}) ∈ Fin → (♯‘(𝑆 ∖ {𝑋})) ∈
ℕ0) |
| 14 | 10, 13 | syl 17 |
. . . . . 6
⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈
ℕ0) |
| 15 | 14 | nn0cnd 12589 |
. . . . 5
⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈ ℂ) |
| 16 | 7 | nn0cnd 12589 |
. . . . 5
⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℂ) |
| 17 | | 1cnd 11256 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℂ) |
| 18 | | undif1 4476 |
. . . . . . . 8
⊢ ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = (𝑆 ∪ {𝑋}) |
| 19 | | ramub1.x |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ 𝑆) |
| 20 | 19 | snssd 4809 |
. . . . . . . . 9
⊢ (𝜑 → {𝑋} ⊆ 𝑆) |
| 21 | | ssequn2 4189 |
. . . . . . . . 9
⊢ ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∪ {𝑋}) = 𝑆) |
| 22 | 20, 21 | sylib 218 |
. . . . . . . 8
⊢ (𝜑 → (𝑆 ∪ {𝑋}) = 𝑆) |
| 23 | 18, 22 | eqtrid 2789 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆) |
| 24 | 23 | fveq2d 6910 |
. . . . . 6
⊢ (𝜑 → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = (♯‘𝑆)) |
| 25 | | neldifsnd 4793 |
. . . . . . 7
⊢ (𝜑 → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) |
| 26 | | hashunsng 14431 |
. . . . . . . 8
⊢ (𝑋 ∈ 𝑆 → (((𝑆 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑆 ∖ {𝑋})) + 1))) |
| 27 | 19, 26 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (((𝑆 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑆 ∖ {𝑋})) + 1))) |
| 28 | 10, 25, 27 | mp2and 699 |
. . . . . 6
⊢ (𝜑 → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑆 ∖ {𝑋})) + 1)) |
| 29 | | ramub1.5 |
. . . . . 6
⊢ (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1)) |
| 30 | 24, 28, 29 | 3eqtr3d 2785 |
. . . . 5
⊢ (𝜑 → ((♯‘(𝑆 ∖ {𝑋})) + 1) = (((𝑀 − 1) Ramsey 𝐺) + 1)) |
| 31 | 15, 16, 17, 30 | addcan2ad 11467 |
. . . 4
⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) = ((𝑀 − 1) Ramsey 𝐺)) |
| 32 | 12, 31 | breqtrrd 5171 |
. . 3
⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ (♯‘(𝑆 ∖ {𝑋}))) |
| 33 | | ramub1.6 |
. . . . . 6
⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅) |
| 34 | 33 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅) |
| 35 | | fveqeq2 6915 |
. . . . . . 7
⊢ (𝑥 = (𝑢 ∪ {𝑋}) → ((♯‘𝑥) = 𝑀 ↔ (♯‘(𝑢 ∪ {𝑋})) = 𝑀)) |
| 36 | 1 | hashbcval 17040 |
. . . . . . . . . . . . . . 15
⊢ (((𝑆 ∖ {𝑋}) ∈ Fin ∧ (𝑀 − 1) ∈ ℕ0)
→ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)}) |
| 37 | 10, 4, 36 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)}) |
| 38 | 37 | eleq2d 2827 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ 𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)})) |
| 39 | | fveqeq2 6915 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝑀 − 1) ↔ (♯‘𝑢) = (𝑀 − 1))) |
| 40 | 39 | elrab 3692 |
. . . . . . . . . . . . 13
⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)} ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (♯‘𝑢) = (𝑀 − 1))) |
| 41 | 38, 40 | bitrdi 287 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (♯‘𝑢) = (𝑀 − 1)))) |
| 42 | 41 | simprbda 498 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋})) |
| 43 | 42 | elpwid 4609 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ (𝑆 ∖ {𝑋})) |
| 44 | 43 | difss2d 4139 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ 𝑆) |
| 45 | 20 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → {𝑋} ⊆ 𝑆) |
| 46 | 44, 45 | unssd 4192 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ⊆ 𝑆) |
| 47 | | vex 3484 |
. . . . . . . . . 10
⊢ 𝑢 ∈ V |
| 48 | | snex 5436 |
. . . . . . . . . 10
⊢ {𝑋} ∈ V |
| 49 | 47, 48 | unex 7764 |
. . . . . . . . 9
⊢ (𝑢 ∪ {𝑋}) ∈ V |
| 50 | 49 | elpw 4604 |
. . . . . . . 8
⊢ ((𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆 ↔ (𝑢 ∪ {𝑋}) ⊆ 𝑆) |
| 51 | 46, 50 | sylibr 234 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆) |
| 52 | 10 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆 ∖ {𝑋}) ∈ Fin) |
| 53 | 52, 43 | ssfid 9301 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ Fin) |
| 54 | | neldifsnd 4793 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) |
| 55 | 43, 54 | ssneldd 3986 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ 𝑢) |
| 56 | 19 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑋 ∈ 𝑆) |
| 57 | | hashunsng 14431 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝑆 → ((𝑢 ∈ Fin ∧ ¬ 𝑋 ∈ 𝑢) → (♯‘(𝑢 ∪ {𝑋})) = ((♯‘𝑢) + 1))) |
| 58 | 56, 57 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((𝑢 ∈ Fin ∧ ¬ 𝑋 ∈ 𝑢) → (♯‘(𝑢 ∪ {𝑋})) = ((♯‘𝑢) + 1))) |
| 59 | 53, 55, 58 | mp2and 699 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘(𝑢 ∪ {𝑋})) = ((♯‘𝑢) + 1)) |
| 60 | 41 | simplbda 499 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘𝑢) = (𝑀 − 1)) |
| 61 | 60 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((♯‘𝑢) + 1) = ((𝑀 − 1) + 1)) |
| 62 | 2 | nncnd 12282 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 63 | | ax-1cn 11213 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
| 64 | | npcan 11517 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑀 −
1) + 1) = 𝑀) |
| 65 | 62, 63, 64 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
| 66 | 65 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀) |
| 67 | 59, 61, 66 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘(𝑢 ∪ {𝑋})) = 𝑀) |
| 68 | 35, 51, 67 | elrabd 3694 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀}) |
| 69 | 2 | nnnn0d 12587 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 70 | 1 | hashbcval 17040 |
. . . . . . . 8
⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ0)
→ (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀}) |
| 71 | 8, 69, 70 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀}) |
| 72 | 71 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀}) |
| 73 | 68, 72 | eleqtrrd 2844 |
. . . . 5
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ (𝑆𝐶𝑀)) |
| 74 | 34, 73 | ffvelcdmd 7105 |
. . . 4
⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝐾‘(𝑢 ∪ {𝑋})) ∈ 𝑅) |
| 75 | | ramub1.h |
. . . 4
⊢ 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋}))) |
| 76 | 74, 75 | fmptd 7134 |
. . 3
⊢ (𝜑 → 𝐻:((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))⟶𝑅) |
| 77 | 1, 4, 5, 6, 7, 10,
32, 76 | rami 17053 |
. 2
⊢ (𝜑 → ∃𝑑 ∈ 𝑅 ∃𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑}))) |
| 78 | 69 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝑀 ∈
ℕ0) |
| 79 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝑅 ∈ Fin) |
| 80 | | ramub1.f |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:𝑅⟶ℕ) |
| 81 | 80 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝐹:𝑅⟶ℕ) |
| 82 | | simprll 779 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝑑 ∈ 𝑅) |
| 83 | 81, 82 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝐹‘𝑑) ∈ ℕ) |
| 84 | | nnm1nn0 12567 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑑) ∈ ℕ → ((𝐹‘𝑑) − 1) ∈
ℕ0) |
| 85 | 83, 84 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → ((𝐹‘𝑑) − 1) ∈
ℕ0) |
| 86 | 85 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → ((𝐹‘𝑑) − 1) ∈
ℕ0) |
| 87 | 81 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → (𝐹‘𝑦) ∈ ℕ) |
| 88 | 87 | nnnn0d 12587 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → (𝐹‘𝑦) ∈
ℕ0) |
| 89 | 86, 88 | ifcld 4572 |
. . . . . . 7
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)) ∈
ℕ0) |
| 90 | | eqid 2737 |
. . . . . . 7
⊢ (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))) = (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))) |
| 91 | 89, 90 | fmptd 7134 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))):𝑅⟶ℕ0) |
| 92 | | equequ2 2025 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑑 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑑)) |
| 93 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑑 → (𝐹‘𝑥) = (𝐹‘𝑑)) |
| 94 | 93 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑑 → ((𝐹‘𝑥) − 1) = ((𝐹‘𝑑) − 1)) |
| 95 | 92, 94 | ifbieq1d 4550 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑑 → if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)) = if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))) |
| 96 | 95 | mpteq2dv 5244 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑑 → (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦))) = (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) |
| 97 | 96 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑥 = 𝑑 → (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)))) = (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))))) |
| 98 | | ramub1.g |
. . . . . . . . 9
⊢ 𝐺 = (𝑥 ∈ 𝑅 ↦ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦))))) |
| 99 | | ovex 7464 |
. . . . . . . . 9
⊢ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ∈ V |
| 100 | 97, 98, 99 | fvmpt 7016 |
. . . . . . . 8
⊢ (𝑑 ∈ 𝑅 → (𝐺‘𝑑) = (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))))) |
| 101 | 82, 100 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) = (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))))) |
| 102 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝐺:𝑅⟶ℕ0) |
| 103 | 102, 82 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) ∈
ℕ0) |
| 104 | 101, 103 | eqeltrrd 2842 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ∈
ℕ0) |
| 105 | | simprlr 780 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) |
| 106 | | simprrl 781 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) ≤ (♯‘𝑤)) |
| 107 | 101, 106 | eqbrtrrd 5167 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ≤ (♯‘𝑤)) |
| 108 | 33 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅) |
| 109 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝑆 ∈ Fin) |
| 110 | 105 | elpwid 4609 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋})) |
| 111 | 110 | difss2d 4139 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → 𝑤 ⊆ 𝑆) |
| 112 | 1 | hashbcss 17042 |
. . . . . . . 8
⊢ ((𝑆 ∈ Fin ∧ 𝑤 ⊆ 𝑆 ∧ 𝑀 ∈ ℕ0) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀)) |
| 113 | 109, 111,
78, 112 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀)) |
| 114 | 108, 113 | fssresd 6775 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝐾 ↾ (𝑤𝐶𝑀)):(𝑤𝐶𝑀)⟶𝑅) |
| 115 | 1, 78, 79, 91, 104, 105, 107, 114 | rami 17053 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → ∃𝑐 ∈ 𝑅 ∃𝑣 ∈ 𝒫 𝑤(((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}))) |
| 116 | | equequ1 2024 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑐 → (𝑦 = 𝑑 ↔ 𝑐 = 𝑑)) |
| 117 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑐 → (𝐹‘𝑦) = (𝐹‘𝑐)) |
| 118 | 116, 117 | ifbieq2d 4552 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑐 → if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐))) |
| 119 | | ovex 7464 |
. . . . . . . . . . . . . 14
⊢ ((𝐹‘𝑑) − 1) ∈ V |
| 120 | | fvex 6919 |
. . . . . . . . . . . . . 14
⊢ (𝐹‘𝑐) ∈ V |
| 121 | 119, 120 | ifex 4576 |
. . . . . . . . . . . . 13
⊢ if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ∈ V |
| 122 | 118, 90, 121 | fvmpt 7016 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ 𝑅 → ((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐))) |
| 123 | 122 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐))) |
| 124 | 123 | breq1d 5153 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → (((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ↔ if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣))) |
| 125 | 124 | anbi1d 631 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) ↔ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) |
| 126 | 2 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑀 ∈ ℕ) |
| 127 | 5 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑅 ∈ Fin) |
| 128 | 80 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐹:𝑅⟶ℕ) |
| 129 | 6 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐺:𝑅⟶ℕ0) |
| 130 | 7 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → ((𝑀 − 1) Ramsey 𝐺) ∈
ℕ0) |
| 131 | 8 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑆 ∈ Fin) |
| 132 | 29 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1)) |
| 133 | 33 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅) |
| 134 | 19 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑋 ∈ 𝑆) |
| 135 | 82 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑑 ∈ 𝑅) |
| 136 | 110 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋})) |
| 137 | 106 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝐺‘𝑑) ≤ (♯‘𝑤)) |
| 138 | | simprrr 782 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})) |
| 139 | 138 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})) |
| 140 | | simprll 779 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑐 ∈ 𝑅) |
| 141 | | simprlr 780 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑣 ∈ 𝒫 𝑤) |
| 142 | 141 | elpwid 4609 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑣 ⊆ 𝑤) |
| 143 | | simprrl 781 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣)) |
| 144 | | simprrr 782 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) |
| 145 | | cnvresima 6250 |
. . . . . . . . . . . . 13
⊢ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) = ((◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀)) |
| 146 | | inss1 4237 |
. . . . . . . . . . . . 13
⊢ ((◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀)) ⊆ (◡𝐾 “ {𝑐}) |
| 147 | 145, 146 | eqsstri 4030 |
. . . . . . . . . . . 12
⊢ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) ⊆ (◡𝐾 “ {𝑐}) |
| 148 | 144, 147 | sstrdi 3996 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑣𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})) |
| 149 | 126, 127,
128, 98, 129, 130, 1, 131, 132, 133, 134, 75, 135, 136, 137, 139, 140, 142, 143, 148 | ramub1lem1 17064 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐}))) |
| 150 | 149 | expr 456 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})))) |
| 151 | 125, 150 | sylbid 240 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})))) |
| 152 | 151 | anassrs 467 |
. . . . . . 7
⊢ ((((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ 𝑐 ∈ 𝑅) ∧ 𝑣 ∈ 𝒫 𝑤) → ((((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})))) |
| 153 | 152 | rexlimdva 3155 |
. . . . . 6
⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) ∧ 𝑐 ∈ 𝑅) → (∃𝑣 ∈ 𝒫 𝑤(((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})))) |
| 154 | 153 | reximdva 3168 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → (∃𝑐 ∈ 𝑅 ∃𝑣 ∈ 𝒫 𝑤(((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})))) |
| 155 | 115, 154 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})))) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐}))) |
| 156 | 155 | expr 456 |
. . 3
⊢ ((𝜑 ∧ (𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋}))) → (((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})))) |
| 157 | 156 | rexlimdvva 3213 |
. 2
⊢ (𝜑 → (∃𝑑 ∈ 𝑅 ∃𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝑑})) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐})))) |
| 158 | 77, 157 | mpd 15 |
1
⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐}))) |