| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ssmapsn.c | . . . . . . . 8
⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑m {𝐴})) | 
| 2 | 1 | sselda 3983 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐵 ↑m {𝐴})) | 
| 3 |  | elmapi 8889 | . . . . . . 7
⊢ (𝑓 ∈ (𝐵 ↑m {𝐴}) → 𝑓:{𝐴}⟶𝐵) | 
| 4 | 2, 3 | syl 17 | . . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓:{𝐴}⟶𝐵) | 
| 5 | 4 | ffnd 6737 | . . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 Fn {𝐴}) | 
| 6 |  | ssmapsn.d | . . . . . . . 8
⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓 | 
| 7 | 6 | a1i 11 | . . . . . . 7
⊢ (𝜑 → 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓) | 
| 8 |  | ovexd 7466 | . . . . . . . . 9
⊢ (𝜑 → (𝐵 ↑m {𝐴}) ∈ V) | 
| 9 | 8, 1 | ssexd 5324 | . . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ V) | 
| 10 |  | rnexg 7924 | . . . . . . . . 9
⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ∈ V) | 
| 11 | 10 | rgen 3063 | . . . . . . . 8
⊢
∀𝑓 ∈
𝐶 ran 𝑓 ∈ V | 
| 12 |  | iunexg 7988 | . . . . . . . 8
⊢ ((𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V) | 
| 13 | 9, 11, 12 | sylancl 586 | . . . . . . 7
⊢ (𝜑 → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V) | 
| 14 | 7, 13 | eqeltrd 2841 | . . . . . 6
⊢ (𝜑 → 𝐷 ∈ V) | 
| 15 | 14 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐷 ∈ V) | 
| 16 |  | ssiun2 5047 | . . . . . . . 8
⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ⊆ ∪
𝑓 ∈ 𝐶 ran 𝑓) | 
| 17 | 16 | adantl 481 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ran 𝑓 ⊆ ∪
𝑓 ∈ 𝐶 ran 𝑓) | 
| 18 |  | ssmapsn.a | . . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| 19 |  | snidg 4660 | . . . . . . . . . 10
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | 
| 20 | 18, 19 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ {𝐴}) | 
| 21 | 20 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐴 ∈ {𝐴}) | 
| 22 | 5, 21 | fnfvelrnd 7102 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ran 𝑓) | 
| 23 | 17, 22 | sseldd 3984 | . . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ∪
𝑓 ∈ 𝐶 ran 𝑓) | 
| 24 | 23, 6 | eleqtrrdi 2852 | . . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ 𝐷) | 
| 25 | 5, 15, 24 | elmapsnd 45209 | . . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐷 ↑m {𝐴})) | 
| 26 | 14 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐷 ∈ V) | 
| 27 |  | snex 5436 | . . . . . . . . 9
⊢ {𝐴} ∈ V | 
| 28 | 27 | a1i 11 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → {𝐴} ∈ V) | 
| 29 |  | simpr 484 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ (𝐷 ↑m {𝐴})) | 
| 30 | 20 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐴 ∈ {𝐴}) | 
| 31 | 26, 28, 29, 30 | fvmap 45203 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ 𝐷) | 
| 32 |  | rneq 5947 | . . . . . . . . 9
⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔) | 
| 33 | 32 | cbviunv 5040 | . . . . . . . 8
⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔 | 
| 34 | 6, 33 | eqtri 2765 | . . . . . . 7
⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔 | 
| 35 | 31, 34 | eleqtrdi 2851 | . . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ ∪
𝑔 ∈ 𝐶 ran 𝑔) | 
| 36 |  | eliun 4995 | . . . . . 6
⊢ ((𝑓‘𝐴) ∈ ∪
𝑔 ∈ 𝐶 ran 𝑔 ↔ ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔) | 
| 37 | 35, 36 | sylib 218 | . . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔) | 
| 38 |  | simp3 1139 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓‘𝐴) ∈ ran 𝑔) | 
| 39 |  | simp1l 1198 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝜑) | 
| 40 | 39, 18 | syl 17 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝐴 ∈ 𝑉) | 
| 41 |  | eqid 2737 | . . . . . . . . 9
⊢ {𝐴} = {𝐴} | 
| 42 |  | simp1r 1199 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷 ↑m {𝐴})) | 
| 43 |  | elmapfn 8905 | . . . . . . . . . 10
⊢ (𝑓 ∈ (𝐷 ↑m {𝐴}) → 𝑓 Fn {𝐴}) | 
| 44 | 42, 43 | syl 17 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴}) | 
| 45 | 1 | sselda 3983 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ (𝐵 ↑m {𝐴})) | 
| 46 |  | elmapfn 8905 | . . . . . . . . . . . 12
⊢ (𝑔 ∈ (𝐵 ↑m {𝐴}) → 𝑔 Fn {𝐴}) | 
| 47 | 45, 46 | syl 17 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 Fn {𝐴}) | 
| 48 | 47 | 3adant3 1133 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴}) | 
| 49 | 48 | 3adant1r 1178 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴}) | 
| 50 | 40, 41, 44, 49 | fsneqrn 45216 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓‘𝐴) ∈ ran 𝑔)) | 
| 51 | 38, 50 | mpbird 257 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔) | 
| 52 |  | simp2 1138 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 ∈ 𝐶) | 
| 53 | 51, 52 | eqeltrd 2841 | . . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ 𝐶) | 
| 54 | 53 | rexlimdv3a 3159 | . . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶)) | 
| 55 | 37, 54 | mpd 15 | . . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ 𝐶) | 
| 56 | 25, 55 | impbida 801 | . . 3
⊢ (𝜑 → (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴}))) | 
| 57 | 56 | alrimiv 1927 | . 2
⊢ (𝜑 → ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴}))) | 
| 58 |  | nfcv 2905 | . . 3
⊢
Ⅎ𝑓𝐶 | 
| 59 |  | ssmapsn.f | . . . 4
⊢
Ⅎ𝑓𝐷 | 
| 60 |  | nfcv 2905 | . . . 4
⊢
Ⅎ𝑓
↑m | 
| 61 |  | nfcv 2905 | . . . 4
⊢
Ⅎ𝑓{𝐴} | 
| 62 | 59, 60, 61 | nfov 7461 | . . 3
⊢
Ⅎ𝑓(𝐷 ↑m {𝐴}) | 
| 63 | 58, 62 | cleqf 2934 | . 2
⊢ (𝐶 = (𝐷 ↑m {𝐴}) ↔ ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴}))) | 
| 64 | 57, 63 | sylibr 234 | 1
⊢ (𝜑 → 𝐶 = (𝐷 ↑m {𝐴})) |