Proof of Theorem modelaxreplem3
| Step | Hyp | Ref
| Expression |
| 1 | | modelaxreplem.1 |
. . 3
⊢ (𝜓 → 𝑥 ⊆ 𝑀) |
| 2 | | modelaxreplem.2 |
. . 3
⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀)) |
| 3 | | modelaxreplem.3 |
. . 3
⊢ (𝜓 → ∅ ∈ 𝑀) |
| 4 | | modelaxreplem.4 |
. . 3
⊢ (𝜓 → 𝑥 ∈ 𝑀) |
| 5 | | modelaxreplem2.5 |
. . 3
⊢
Ⅎ𝑤𝜓 |
| 6 | | modelaxreplem2.6 |
. . 3
⊢
Ⅎ𝑧𝜓 |
| 7 | | modelaxreplem2.7 |
. . 3
⊢
Ⅎ𝑧𝐹 |
| 8 | | modelaxreplem2.8 |
. . 3
⊢ 𝐹 = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} |
| 9 | | modelaxreplem2.9 |
. . 3
⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦))) |
| 10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | modelaxreplem2 44987 |
. 2
⊢ (𝜓 → ran 𝐹 ∈ 𝑀) |
| 11 | 1 | sseld 3981 |
. . . . . . . . . 10
⊢ (𝜓 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑀)) |
| 12 | 11 | pm4.71rd 562 |
. . . . . . . . 9
⊢ (𝜓 → (𝑤 ∈ 𝑥 ↔ (𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥))) |
| 13 | 12 | anbi1d 631 |
. . . . . . . 8
⊢ (𝜓 → ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))) |
| 14 | | an12 645 |
. . . . . . . . 9
⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑))) |
| 15 | | anass 468 |
. . . . . . . . . 10
⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑) ↔ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| 16 | 15 | anbi2i 623 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝑀 ∧ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 17 | 14, 16 | bitri 275 |
. . . . . . . 8
⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 18 | 13, 17 | bitrdi 287 |
. . . . . . 7
⊢ (𝜓 → ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))) |
| 19 | 5, 18 | exbid 2223 |
. . . . . 6
⊢ (𝜓 → (∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))) |
| 20 | 8 | rneqi 5946 |
. . . . . . . 8
⊢ ran 𝐹 = ran {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} |
| 21 | | rnopab 5963 |
. . . . . . . 8
⊢ ran
{〈𝑤, 𝑧〉 ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} |
| 22 | 20, 21 | eqtri 2764 |
. . . . . . 7
⊢ ran 𝐹 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} |
| 23 | 22 | eqabri 2884 |
. . . . . 6
⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))) |
| 24 | | df-rex 3070 |
. . . . . . . 8
⊢
(∃𝑤 ∈
𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| 25 | 24 | anbi2i 623 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 26 | | 19.42v 1953 |
. . . . . . 7
⊢
(∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 27 | 25, 26 | bitr4i 278 |
. . . . . 6
⊢ ((𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 28 | 19, 23, 27 | 3bitr4g 314 |
. . . . 5
⊢ (𝜓 → (𝑧 ∈ ran 𝐹 ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 29 | 28 | baibd 539 |
. . . 4
⊢ ((𝜓 ∧ 𝑧 ∈ 𝑀) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| 30 | 6, 29 | ralrimia 3257 |
. . 3
⊢ (𝜓 → ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| 31 | 7 | nfrn 5961 |
. . . . . 6
⊢
Ⅎ𝑧ran
𝐹 |
| 32 | | sbcralt 3871 |
. . . . . 6
⊢ ((ran
𝐹 ∈ 𝑀 ∧ Ⅎ𝑧ran 𝐹) → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 [ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 33 | 31, 32 | mpan2 691 |
. . . . 5
⊢ (ran
𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 [ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 34 | 31 | nfel1 2921 |
. . . . . 6
⊢
Ⅎ𝑧ran 𝐹 ∈ 𝑀 |
| 35 | | sbcbig 3839 |
. . . . . . 7
⊢ (ran
𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ [ran 𝐹 / 𝑦]∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 36 | | sbcel2gv 3856 |
. . . . . . . 8
⊢ (ran
𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ran 𝐹)) |
| 37 | | nfcv 2904 |
. . . . . . . . . 10
⊢
Ⅎ𝑦𝑀 |
| 38 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦 𝑤 ∈ 𝑥 |
| 39 | | nfa1 2151 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦∀𝑦𝜑 |
| 40 | 38, 39 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) |
| 41 | 37, 40 | nfrexw 3312 |
. . . . . . . . 9
⊢
Ⅎ𝑦∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) |
| 42 | 41 | sbcgf 3860 |
. . . . . . . 8
⊢ (ran
𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| 43 | 36, 42 | bibi12d 345 |
. . . . . . 7
⊢ (ran
𝐹 ∈ 𝑀 → (([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ [ran 𝐹 / 𝑦]∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 44 | 35, 43 | bitrd 279 |
. . . . . 6
⊢ (ran
𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 45 | 34, 44 | ralbid 3272 |
. . . . 5
⊢ (ran
𝐹 ∈ 𝑀 → (∀𝑧 ∈ 𝑀 [ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 46 | 33, 45 | bitrd 279 |
. . . 4
⊢ (ran
𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 47 | 10, 46 | syl 17 |
. . 3
⊢ (𝜓 → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 48 | 30, 47 | mpbird 257 |
. 2
⊢ (𝜓 → [ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| 49 | 10, 48 | rspesbcd 44951 |
1
⊢ (𝜓 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |