| Step | Hyp | Ref
| Expression |
| 1 | | smfpimbor1lem2.p |
. 2
⊢ 𝑃 = (◡𝐹 “ 𝐸) |
| 2 | | smfpimbor1lem2.j |
. . . . . . . 8
⊢ 𝐽 = (topGen‘ran
(,)) |
| 3 | | retop 24782 |
. . . . . . . 8
⊢
(topGen‘ran (,)) ∈ Top |
| 4 | 2, 3 | eqeltri 2837 |
. . . . . . 7
⊢ 𝐽 ∈ Top |
| 5 | 4 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ Top) |
| 6 | | smfpimbor1lem2.b |
. . . . . 6
⊢ 𝐵 = (SalGen‘𝐽) |
| 7 | | smfpimbor1lem2.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 8 | | smfpimbor1lem2.f |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
| 9 | | smfpimbor1lem2.a |
. . . . . . 7
⊢ 𝐷 = dom 𝐹 |
| 10 | | smfpimbor1lem2.t |
. . . . . . 7
⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} |
| 11 | 7, 8, 9, 10 | smfresal 46803 |
. . . . . 6
⊢ (𝜑 → 𝑇 ∈ SAlg) |
| 12 | 7 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑆 ∈ SAlg) |
| 13 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹 ∈ (SMblFn‘𝑆)) |
| 14 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽) |
| 15 | 12, 13, 9, 2, 14, 10 | smfpimbor1lem1 46813 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝑇) |
| 16 | 15 | ssd 45085 |
. . . . . 6
⊢ (𝜑 → 𝐽 ⊆ 𝑇) |
| 17 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑒𝑥 |
| 18 | | nfrab1 3457 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑒{𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} |
| 19 | 10, 18 | nfcxfr 2903 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑒𝑇 |
| 20 | 17, 19 | eluni2f 45108 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ∪ 𝑇
↔ ∃𝑒 ∈
𝑇 𝑥 ∈ 𝑒) |
| 21 | 20 | biimpi 216 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∪ 𝑇
→ ∃𝑒 ∈
𝑇 𝑥 ∈ 𝑒) |
| 22 | 19 | nfuni 4914 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑒∪ 𝑇 |
| 23 | 17, 22 | nfel 2920 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑒 𝑥 ∈ ∪ 𝑇 |
| 24 | | nfv 1914 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑒 𝑥 ∈ ℝ |
| 25 | 10 | eleq2i 2833 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑒 ∈ 𝑇 ↔ 𝑒 ∈ {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)}) |
| 26 | 25 | biimpi 216 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑒 ∈ 𝑇 → 𝑒 ∈ {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)}) |
| 27 | | rabidim1 3459 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑒 ∈ {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} → 𝑒 ∈ 𝒫 ℝ) |
| 28 | 26, 27 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑒 ∈ 𝑇 → 𝑒 ∈ 𝒫 ℝ) |
| 29 | | elpwi 4607 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑒 ∈ 𝒫 ℝ →
𝑒 ⊆
ℝ) |
| 30 | 28, 29 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑒 ∈ 𝑇 → 𝑒 ⊆ ℝ) |
| 31 | 30 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑒 ∈ 𝑇 ∧ 𝑥 ∈ 𝑒) → 𝑒 ⊆ ℝ) |
| 32 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑒 ∈ 𝑇 ∧ 𝑥 ∈ 𝑒) → 𝑥 ∈ 𝑒) |
| 33 | 31, 32 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ ((𝑒 ∈ 𝑇 ∧ 𝑥 ∈ 𝑒) → 𝑥 ∈ ℝ) |
| 34 | 33 | ex 412 |
. . . . . . . . . . . . . 14
⊢ (𝑒 ∈ 𝑇 → (𝑥 ∈ 𝑒 → 𝑥 ∈ ℝ)) |
| 35 | 34 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ∪ 𝑇
→ (𝑒 ∈ 𝑇 → (𝑥 ∈ 𝑒 → 𝑥 ∈ ℝ))) |
| 36 | 23, 24, 35 | rexlimd 3266 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∪ 𝑇
→ (∃𝑒 ∈
𝑇 𝑥 ∈ 𝑒 → 𝑥 ∈ ℝ)) |
| 37 | 21, 36 | mpd 15 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ 𝑇
→ 𝑥 ∈
ℝ) |
| 38 | 37 | rgen 3063 |
. . . . . . . . . 10
⊢
∀𝑥 ∈
∪ 𝑇𝑥 ∈ ℝ |
| 39 | | dfss3 3972 |
. . . . . . . . . 10
⊢ (∪ 𝑇
⊆ ℝ ↔ ∀𝑥 ∈ ∪ 𝑇𝑥 ∈ ℝ) |
| 40 | 38, 39 | mpbir 231 |
. . . . . . . . 9
⊢ ∪ 𝑇
⊆ ℝ |
| 41 | 40 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ∪ 𝑇
⊆ ℝ) |
| 42 | | uniretop 24783 |
. . . . . . . . . . . 12
⊢ ℝ =
∪ (topGen‘ran (,)) |
| 43 | 2 | eqcomi 2746 |
. . . . . . . . . . . . 13
⊢
(topGen‘ran (,)) = 𝐽 |
| 44 | 43 | unieqi 4919 |
. . . . . . . . . . . 12
⊢ ∪ (topGen‘ran (,)) = ∪
𝐽 |
| 45 | 42, 44 | eqtr2i 2766 |
. . . . . . . . . . 11
⊢ ∪ 𝐽 =
ℝ |
| 46 | 45 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝐽 =
ℝ) |
| 47 | 46 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝜑 → ℝ = ∪ 𝐽) |
| 48 | 16 | unissd 4917 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝐽
⊆ ∪ 𝑇) |
| 49 | 47, 48 | eqsstrd 4018 |
. . . . . . . 8
⊢ (𝜑 → ℝ ⊆ ∪ 𝑇) |
| 50 | 41, 49 | eqssd 4001 |
. . . . . . 7
⊢ (𝜑 → ∪ 𝑇 =
ℝ) |
| 51 | 50, 46 | eqtr4d 2780 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑇 =
∪ 𝐽) |
| 52 | 5, 6, 11, 16, 51 | salgenss 46351 |
. . . . 5
⊢ (𝜑 → 𝐵 ⊆ 𝑇) |
| 53 | | smfpimbor1lem2.e |
. . . . 5
⊢ (𝜑 → 𝐸 ∈ 𝐵) |
| 54 | 52, 53 | sseldd 3984 |
. . . 4
⊢ (𝜑 → 𝐸 ∈ 𝑇) |
| 55 | | imaeq2 6074 |
. . . . . 6
⊢ (𝑒 = 𝐸 → (◡𝐹 “ 𝑒) = (◡𝐹 “ 𝐸)) |
| 56 | 55 | eleq1d 2826 |
. . . . 5
⊢ (𝑒 = 𝐸 → ((◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ 𝐸) ∈ (𝑆 ↾t 𝐷))) |
| 57 | 56, 10 | elrab2 3695 |
. . . 4
⊢ (𝐸 ∈ 𝑇 ↔ (𝐸 ∈ 𝒫 ℝ ∧ (◡𝐹 “ 𝐸) ∈ (𝑆 ↾t 𝐷))) |
| 58 | 54, 57 | sylib 218 |
. . 3
⊢ (𝜑 → (𝐸 ∈ 𝒫 ℝ ∧ (◡𝐹 “ 𝐸) ∈ (𝑆 ↾t 𝐷))) |
| 59 | 58 | simprd 495 |
. 2
⊢ (𝜑 → (◡𝐹 “ 𝐸) ∈ (𝑆 ↾t 𝐷)) |
| 60 | 1, 59 | eqeltrid 2845 |
1
⊢ (𝜑 → 𝑃 ∈ (𝑆 ↾t 𝐷)) |