| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq2 7439 | . . 3
⊢ (𝐵 = ∅ → (𝐴 ↑m 𝐵) = (𝐴 ↑m
∅)) | 
| 2 |  | breq2 5147 | . . . . 5
⊢ (𝐵 = ∅ → (𝑥 ≈ 𝐵 ↔ 𝑥 ≈ ∅)) | 
| 3 | 2 | anbi2d 630 | . . . 4
⊢ (𝐵 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))) | 
| 4 | 3 | abbidv 2808 | . . 3
⊢ (𝐵 = ∅ → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}) | 
| 5 | 1, 4 | breq12d 5156 | . 2
⊢ (𝐵 = ∅ → ((𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ↔ (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})) | 
| 6 |  | simpl2 1193 | . . . . . . . . . 10
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≼ 𝐴) | 
| 7 |  | reldom 8991 | . . . . . . . . . . 11
⊢ Rel
≼ | 
| 8 | 7 | brrelex1i 5741 | . . . . . . . . . 10
⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V) | 
| 9 | 6, 8 | syl 17 | . . . . . . . . 9
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V) | 
| 10 | 7 | brrelex2i 5742 | . . . . . . . . . 10
⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V) | 
| 11 | 6, 10 | syl 17 | . . . . . . . . 9
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V) | 
| 12 |  | xpcomeng 9104 | . . . . . . . . 9
⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) | 
| 13 | 9, 11, 12 | syl2anc 584 | . . . . . . . 8
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) | 
| 14 |  | simpl3 1194 | . . . . . . . . . 10
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ∈ dom card) | 
| 15 |  | simpr 484 | . . . . . . . . . . 11
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅) | 
| 16 |  | mapdom3 9189 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵)) | 
| 17 | 11, 9, 15, 16 | syl3anc 1373 | . . . . . . . . . 10
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵)) | 
| 18 |  | numdom 10078 | . . . . . . . . . 10
⊢ (((𝐴 ↑m 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴 ↑m 𝐵)) → 𝐴 ∈ dom card) | 
| 19 | 14, 17, 18 | syl2anc 584 | . . . . . . . . 9
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card) | 
| 20 |  | simpl1 1192 | . . . . . . . . 9
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴) | 
| 21 |  | infxpabs 10251 | . . . . . . . . 9
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴) | 
| 22 | 19, 20, 15, 6, 21 | syl22anc 839 | . . . . . . . 8
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴) | 
| 23 |  | entr 9046 | . . . . . . . 8
⊢ (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴) | 
| 24 | 13, 22, 23 | syl2anc 584 | . . . . . . 7
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴) | 
| 25 |  | ssenen 9191 | . . . . . . 7
⊢ ((𝐵 × 𝐴) ≈ 𝐴 → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | 
| 26 | 24, 25 | syl 17 | . . . . . 6
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | 
| 27 |  | relen 8990 | . . . . . . 7
⊢ Rel
≈ | 
| 28 | 27 | brrelex1i 5741 | . . . . . 6
⊢ ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V) | 
| 29 | 26, 28 | syl 17 | . . . . 5
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V) | 
| 30 |  | abid2 2879 | . . . . . 6
⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} = (𝐴 ↑m 𝐵) | 
| 31 |  | elmapi 8889 | . . . . . . . 8
⊢ (𝑥 ∈ (𝐴 ↑m 𝐵) → 𝑥:𝐵⟶𝐴) | 
| 32 |  | fssxp 6763 | . . . . . . . . 9
⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ⊆ (𝐵 × 𝐴)) | 
| 33 |  | ffun 6739 | . . . . . . . . . . 11
⊢ (𝑥:𝐵⟶𝐴 → Fun 𝑥) | 
| 34 |  | vex 3484 | . . . . . . . . . . . 12
⊢ 𝑥 ∈ V | 
| 35 | 34 | fundmen 9071 | . . . . . . . . . . 11
⊢ (Fun
𝑥 → dom 𝑥 ≈ 𝑥) | 
| 36 |  | ensym 9043 | . . . . . . . . . . 11
⊢ (dom
𝑥 ≈ 𝑥 → 𝑥 ≈ dom 𝑥) | 
| 37 | 33, 35, 36 | 3syl 18 | . . . . . . . . . 10
⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ dom 𝑥) | 
| 38 |  | fdm 6745 | . . . . . . . . . 10
⊢ (𝑥:𝐵⟶𝐴 → dom 𝑥 = 𝐵) | 
| 39 | 37, 38 | breqtrd 5169 | . . . . . . . . 9
⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ 𝐵) | 
| 40 | 32, 39 | jca 511 | . . . . . . . 8
⊢ (𝑥:𝐵⟶𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)) | 
| 41 | 31, 40 | syl 17 | . . . . . . 7
⊢ (𝑥 ∈ (𝐴 ↑m 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)) | 
| 42 | 41 | ss2abi 4067 | . . . . . 6
⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} | 
| 43 | 30, 42 | eqsstrri 4031 | . . . . 5
⊢ (𝐴 ↑m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} | 
| 44 |  | ssdomg 9040 | . . . . 5
⊢ ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V → ((𝐴 ↑m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)})) | 
| 45 | 29, 43, 44 | mpisyl 21 | . . . 4
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}) | 
| 46 |  | domentr 9053 | . . . 4
⊢ (((𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | 
| 47 | 45, 26, 46 | syl2anc 584 | . . 3
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | 
| 48 |  | ovex 7464 | . . . . . . 7
⊢ (𝐴 ↑m 𝐵) ∈ V | 
| 49 | 48 | mptex 7243 | . . . . . 6
⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V | 
| 50 | 49 | rnex 7932 | . . . . 5
⊢ ran
(𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V | 
| 51 |  | ensym 9043 | . . . . . . . . . . . 12
⊢ (𝑥 ≈ 𝐵 → 𝐵 ≈ 𝑥) | 
| 52 | 51 | ad2antll 729 | . . . . . . . . . . 11
⊢
((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ≈ 𝑥) | 
| 53 |  | bren 8995 | . . . . . . . . . . 11
⊢ (𝐵 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑥) | 
| 54 | 52, 53 | sylib 218 | . . . . . . . . . 10
⊢
((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑥) | 
| 55 |  | f1of 6848 | . . . . . . . . . . . . . . . 16
⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵⟶𝑥) | 
| 56 | 55 | adantl 481 | . . . . . . . . . . . . . . 15
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑓:𝐵⟶𝑥) | 
| 57 |  | simplrl 777 | . . . . . . . . . . . . . . 15
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑥 ⊆ 𝐴) | 
| 58 | 56, 57 | fssd 6753 | . . . . . . . . . . . . . 14
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑓:𝐵⟶𝐴) | 
| 59 | 11, 9 | elmapd 8880 | . . . . . . . . . . . . . . 15
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐴)) | 
| 60 | 59 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐴)) | 
| 61 | 58, 60 | mpbird 257 | . . . . . . . . . . . . 13
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑓 ∈ (𝐴 ↑m 𝐵)) | 
| 62 |  | f1ofo 6855 | . . . . . . . . . . . . . . . 16
⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵–onto→𝑥) | 
| 63 |  | forn 6823 | . . . . . . . . . . . . . . . 16
⊢ (𝑓:𝐵–onto→𝑥 → ran 𝑓 = 𝑥) | 
| 64 | 62, 63 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝑓:𝐵–1-1-onto→𝑥 → ran 𝑓 = 𝑥) | 
| 65 | 64 | adantl 481 | . . . . . . . . . . . . . 14
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → ran 𝑓 = 𝑥) | 
| 66 | 65 | eqcomd 2743 | . . . . . . . . . . . . 13
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑥 = ran 𝑓) | 
| 67 | 61, 66 | jca 511 | . . . . . . . . . . . 12
⊢
(((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → (𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓)) | 
| 68 | 67 | ex 412 | . . . . . . . . . . 11
⊢
((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → (𝑓:𝐵–1-1-onto→𝑥 → (𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓))) | 
| 69 | 68 | eximdv 1917 | . . . . . . . . . 10
⊢
((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → (∃𝑓 𝑓:𝐵–1-1-onto→𝑥 → ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓))) | 
| 70 | 54, 69 | mpd 15 | . . . . . . . . 9
⊢
((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓)) | 
| 71 |  | df-rex 3071 | . . . . . . . . 9
⊢
(∃𝑓 ∈
(𝐴 ↑m 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓)) | 
| 72 | 70, 71 | sylibr 234 | . . . . . . . 8
⊢
((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓) | 
| 73 | 72 | ex 412 | . . . . . . 7
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) → ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓)) | 
| 74 | 73 | ss2abdv 4066 | . . . . . 6
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓}) | 
| 75 |  | eqid 2737 | . . . . . . 7
⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) | 
| 76 | 75 | rnmpt 5968 | . . . . . 6
⊢ ran
(𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓} | 
| 77 | 74, 76 | sseqtrrdi 4025 | . . . . 5
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ⊆ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)) | 
| 78 |  | ssdomg 9040 | . . . . 5
⊢ (ran
(𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V → ({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ⊆ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓))) | 
| 79 | 50, 77, 78 | mpsyl 68 | . . . 4
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)) | 
| 80 |  | vex 3484 | . . . . . . . . 9
⊢ 𝑓 ∈ V | 
| 81 | 80 | rnex 7932 | . . . . . . . 8
⊢ ran 𝑓 ∈ V | 
| 82 | 81 | rgenw 3065 | . . . . . . 7
⊢
∀𝑓 ∈
(𝐴 ↑m 𝐵)ran 𝑓 ∈ V | 
| 83 | 75 | fnmpt 6708 | . . . . . . 7
⊢
(∀𝑓 ∈
(𝐴 ↑m 𝐵)ran 𝑓 ∈ V → (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑m 𝐵)) | 
| 84 | 82, 83 | mp1i 13 | . . . . . 6
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑m 𝐵)) | 
| 85 |  | dffn4 6826 | . . . . . 6
⊢ ((𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑m 𝐵) ↔ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)–onto→ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)) | 
| 86 | 84, 85 | sylib 218 | . . . . 5
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)–onto→ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)) | 
| 87 |  | fodomnum 10097 | . . . . 5
⊢ ((𝐴 ↑m 𝐵) ∈ dom card → ((𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)–onto→ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) → ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ≼ (𝐴 ↑m 𝐵))) | 
| 88 | 14, 86, 87 | sylc 65 | . . . 4
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ≼ (𝐴 ↑m 𝐵)) | 
| 89 |  | domtr 9047 | . . . 4
⊢ (({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ≼ (𝐴 ↑m 𝐵)) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵)) | 
| 90 | 79, 88, 89 | syl2anc 584 | . . 3
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵)) | 
| 91 |  | sbth 9133 | . . 3
⊢ (((𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵)) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | 
| 92 | 47, 90, 91 | syl2anc 584 | . 2
⊢
(((ω ≼ 𝐴
∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | 
| 93 | 7 | brrelex2i 5742 | . . . . 5
⊢ (ω
≼ 𝐴 → 𝐴 ∈ V) | 
| 94 | 93 | 3ad2ant1 1134 | . . . 4
⊢ ((ω
≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → 𝐴 ∈ V) | 
| 95 |  | map0e 8922 | . . . 4
⊢ (𝐴 ∈ V → (𝐴 ↑m ∅) =
1o) | 
| 96 | 94, 95 | syl 17 | . . 3
⊢ ((ω
≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m ∅) =
1o) | 
| 97 |  | 1oex 8516 | . . . . 5
⊢
1o ∈ V | 
| 98 | 97 | enref 9025 | . . . 4
⊢
1o ≈ 1o | 
| 99 |  | df-sn 4627 | . . . . 5
⊢ {∅}
= {𝑥 ∣ 𝑥 = ∅} | 
| 100 |  | df1o2 8513 | . . . . 5
⊢
1o = {∅} | 
| 101 |  | en0 9058 | . . . . . . . 8
⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅) | 
| 102 | 101 | anbi2i 623 | . . . . . . 7
⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅)) | 
| 103 |  | 0ss 4400 | . . . . . . . . 9
⊢ ∅
⊆ 𝐴 | 
| 104 |  | sseq1 4009 | . . . . . . . . 9
⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) | 
| 105 | 103, 104 | mpbiri 258 | . . . . . . . 8
⊢ (𝑥 = ∅ → 𝑥 ⊆ 𝐴) | 
| 106 | 105 | pm4.71ri 560 | . . . . . . 7
⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅)) | 
| 107 | 102, 106 | bitr4i 278 | . . . . . 6
⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ 𝑥 = ∅) | 
| 108 | 107 | abbii 2809 | . . . . 5
⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = {𝑥 ∣ 𝑥 = ∅} | 
| 109 | 99, 100, 108 | 3eqtr4ri 2776 | . . . 4
⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} =
1o | 
| 110 | 98, 109 | breqtrri 5170 | . . 3
⊢
1o ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} | 
| 111 | 96, 110 | eqbrtrdi 5182 | . 2
⊢ ((ω
≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}) | 
| 112 | 5, 92, 111 | pm2.61ne 3027 | 1
⊢ ((ω
≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) |