Proof of Theorem ffsrn
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ffsrn.1 | . . . . . 6
⊢ (𝜑 → Fun 𝐹) | 
| 2 |  | dfdm4 5905 | . . . . . . 7
⊢ dom 𝐹 = ran ◡𝐹 | 
| 3 |  | dfrn4 6221 | . . . . . . 7
⊢ ran ◡𝐹 = (◡𝐹 “ V) | 
| 4 | 2, 3 | eqtri 2764 | . . . . . 6
⊢ dom 𝐹 = (◡𝐹 “ V) | 
| 5 |  | df-fn 6563 | . . . . . . 7
⊢ (𝐹 Fn (◡𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V))) | 
| 6 |  | fnresdm 6686 | . . . . . . 7
⊢ (𝐹 Fn (◡𝐹 “ V) → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹) | 
| 7 | 5, 6 | sylbir 235 | . . . . . 6
⊢ ((Fun
𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)) → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹) | 
| 8 | 1, 4, 7 | sylancl 586 | . . . . 5
⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹) | 
| 9 |  | imaundi 6168 | . . . . . . 7
⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍})) | 
| 10 | 9 | reseq2i 5993 | . . . . . 6
⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))) | 
| 11 |  | undif1 4475 | . . . . . . . . 9
⊢ ((V
∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍}) | 
| 12 |  | ssv 4007 | . . . . . . . . . 10
⊢ {𝑍} ⊆ V | 
| 13 |  | ssequn2 4188 | . . . . . . . . . 10
⊢ ({𝑍} ⊆ V ↔ (V ∪
{𝑍}) = V) | 
| 14 | 12, 13 | mpbi 230 | . . . . . . . . 9
⊢ (V ∪
{𝑍}) = V | 
| 15 | 11, 14 | eqtri 2764 | . . . . . . . 8
⊢ ((V
∖ {𝑍}) ∪ {𝑍}) = V | 
| 16 | 15 | imaeq2i 6075 | . . . . . . 7
⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (◡𝐹 “ V) | 
| 17 | 16 | reseq2i 5993 | . . . . . 6
⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (◡𝐹 “ V)) | 
| 18 |  | resundi 6010 | . . . . . 6
⊢ (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) | 
| 19 | 10, 17, 18 | 3eqtr3i 2772 | . . . . 5
⊢ (𝐹 ↾ (◡𝐹 “ V)) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) | 
| 20 | 8, 19 | eqtr3di 2791 | . . . 4
⊢ (𝜑 → 𝐹 = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))) | 
| 21 | 20 | rneqd 5948 | . . 3
⊢ (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))) | 
| 22 |  | rnun 6164 | . . 3
⊢ ran
((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))) | 
| 23 | 21, 22 | eqtrdi 2792 | . 2
⊢ (𝜑 → ran 𝐹 = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍})))) | 
| 24 |  | ffsrn.0 | . . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝑉) | 
| 25 |  | ffsrn.z | . . . . . 6
⊢ (𝜑 → 𝑍 ∈ 𝑊) | 
| 26 |  | suppimacnv 8200 | . . . . . 6
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | 
| 27 | 24, 25, 26 | syl2anc 584 | . . . . 5
⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | 
| 28 |  | ffsrn.2 | . . . . 5
⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) | 
| 29 | 27, 28 | eqeltrrd 2841 | . . . 4
⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) | 
| 30 |  | cnvexg 7947 | . . . . . 6
⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V) | 
| 31 |  | imaexg 7936 | . . . . . 6
⊢ (◡𝐹 ∈ V → (◡𝐹 “ (V ∖ {𝑍})) ∈ V) | 
| 32 | 24, 30, 31 | 3syl 18 | . . . . 5
⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ V) | 
| 33 |  | cnvimass 6099 | . . . . . . 7
⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹 | 
| 34 |  | fores 6829 | . . . . . . 7
⊢ ((Fun
𝐹 ∧ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹) → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))):(◡𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (◡𝐹 “ (V ∖ {𝑍})))) | 
| 35 | 1, 33, 34 | sylancl 586 | . . . . . 6
⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))):(◡𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (◡𝐹 “ (V ∖ {𝑍})))) | 
| 36 |  | fofn 6821 | . . . . . 6
⊢ ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))):(◡𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (◡𝐹 “ (V ∖ {𝑍}))) → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) Fn (◡𝐹 “ (V ∖ {𝑍}))) | 
| 37 | 35, 36 | syl 17 | . . . . 5
⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) Fn (◡𝐹 “ (V ∖ {𝑍}))) | 
| 38 |  | fnrndomg 10577 | . . . . 5
⊢ ((◡𝐹 “ (V ∖ {𝑍})) ∈ V → ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) Fn (◡𝐹 “ (V ∖ {𝑍})) → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ≼ (◡𝐹 “ (V ∖ {𝑍})))) | 
| 39 | 32, 37, 38 | sylc 65 | . . . 4
⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ≼ (◡𝐹 “ (V ∖ {𝑍}))) | 
| 40 |  | domfi 9230 | . . . 4
⊢ (((◡𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ≼ (◡𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) | 
| 41 | 29, 39, 40 | syl2anc 584 | . . 3
⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) | 
| 42 |  | snfi 9084 | . . . 4
⊢ {𝑍} ∈ Fin | 
| 43 |  | df-ima 5697 | . . . . . 6
⊢ (𝐹 “ (◡𝐹 “ {𝑍})) = ran (𝐹 ↾ (◡𝐹 “ {𝑍})) | 
| 44 |  | funimacnv 6646 | . . . . . . 7
⊢ (Fun
𝐹 → (𝐹 “ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹)) | 
| 45 | 1, 44 | syl 17 | . . . . . 6
⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹)) | 
| 46 | 43, 45 | eqtr3id 2790 | . . . . 5
⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹)) | 
| 47 |  | inss1 4236 | . . . . 5
⊢ ({𝑍} ∩ ran 𝐹) ⊆ {𝑍} | 
| 48 | 46, 47 | eqsstrdi 4027 | . . . 4
⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍}) | 
| 49 |  | ssfi 9214 | . . . 4
⊢ (({𝑍} ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍}) → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin) | 
| 50 | 42, 48, 49 | sylancr 587 | . . 3
⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin) | 
| 51 |  | unfi 9212 | . . 3
⊢ ((ran
(𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin) → (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))) ∈ Fin) | 
| 52 | 41, 50, 51 | syl2anc 584 | . 2
⊢ (𝜑 → (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))) ∈ Fin) | 
| 53 | 23, 52 | eqeltrd 2840 | 1
⊢ (𝜑 → ran 𝐹 ∈ Fin) |