| Step | Hyp | Ref
| Expression |
| 1 | | modelaxrep.1 |
. 2
⊢ (𝜓 → Tr 𝑀) |
| 2 | | modelaxrep.2 |
. . 3
⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀)) |
| 3 | | funeq 6584 |
. . . . . 6
⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔)) |
| 4 | | dmeq 5912 |
. . . . . . 7
⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔) |
| 5 | 4 | eleq1d 2825 |
. . . . . 6
⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀)) |
| 6 | | rneq 5945 |
. . . . . . 7
⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔) |
| 7 | 6 | sseq1d 4014 |
. . . . . 6
⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀)) |
| 8 | 3, 5, 7 | 3anbi123d 1438 |
. . . . 5
⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀))) |
| 9 | 6 | eleq1d 2825 |
. . . . 5
⊢ (𝑓 = 𝑔 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝑔 ∈ 𝑀)) |
| 10 | 8, 9 | imbi12d 344 |
. . . 4
⊢ (𝑓 = 𝑔 → (((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) ↔ ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))) |
| 11 | 10 | cbvalvw 2035 |
. . 3
⊢
(∀𝑓((Fun
𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) ↔ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) |
| 12 | 2, 11 | sylib 218 |
. 2
⊢ (𝜓 → ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) |
| 13 | | modelaxrep.3 |
. 2
⊢ (𝜓 → ∅ ∈ 𝑀) |
| 14 | | trss 5268 |
. . . . . . 7
⊢ (Tr 𝑀 → (𝑥 ∈ 𝑀 → 𝑥 ⊆ 𝑀)) |
| 15 | 14 | imp 406 |
. . . . . 6
⊢ ((Tr
𝑀 ∧ 𝑥 ∈ 𝑀) → 𝑥 ⊆ 𝑀) |
| 16 | 15 | ad5ant14 758 |
. . . . 5
⊢ (((((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → 𝑥 ⊆ 𝑀) |
| 17 | | simp-4r 784 |
. . . . 5
⊢ (((((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) |
| 18 | | simpllr 776 |
. . . . 5
⊢ (((((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → ∅ ∈ 𝑀) |
| 19 | | simplr 769 |
. . . . 5
⊢ (((((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → 𝑥 ∈ 𝑀) |
| 20 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑤(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) |
| 21 | | nfra1 3283 |
. . . . . 6
⊢
Ⅎ𝑤∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) |
| 22 | 20, 21 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑤((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) |
| 23 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑧(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) |
| 24 | | nfcv 2904 |
. . . . . . 7
⊢
Ⅎ𝑧𝑀 |
| 25 | | nfra1 3283 |
. . . . . . . 8
⊢
Ⅎ𝑧∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) |
| 26 | 24, 25 | nfrexw 3312 |
. . . . . . 7
⊢
Ⅎ𝑧∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) |
| 27 | 24, 26 | nfralw 3310 |
. . . . . 6
⊢
Ⅎ𝑧∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) |
| 28 | 23, 27 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑧((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) |
| 29 | | nfopab2 5212 |
. . . . 5
⊢
Ⅎ𝑧{〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} |
| 30 | | eqid 2736 |
. . . . 5
⊢
{〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} |
| 31 | | rsp 3246 |
. . . . . 6
⊢
(∀𝑤 ∈
𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦))) |
| 32 | 31 | adantl 481 |
. . . . 5
⊢ (((((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦))) |
| 33 | 16, 17, 18, 19, 22, 28, 29, 30, 32 | modelaxreplem3 44973 |
. . . 4
⊢ (((((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| 34 | 33 | ex 412 |
. . 3
⊢ ((((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) → (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 35 | 34 | ralrimiva 3145 |
. 2
⊢ (((Tr
𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 36 | 1, 12, 13, 35 | syl21anc 838 |
1
⊢ (𝜓 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |