Step |
Hyp |
Ref |
Expression |
1 |
|
unirnmapsn.A |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
unirnmapsn.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
unirnmapsn.C |
⊢ 𝐶 = { 𝐴 } |
4 |
|
unirnmapsn.x |
⊢ ( 𝜑 → 𝑋 ⊆ ( 𝐵 ↑m 𝐶 ) ) |
5 |
|
snex |
⊢ { 𝐴 } ∈ V |
6 |
3 5
|
eqeltri |
⊢ 𝐶 ∈ V |
7 |
6
|
a1i |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
8 |
7 4
|
unirnmap |
⊢ ( 𝜑 → 𝑋 ⊆ ( ran ∪ 𝑋 ↑m 𝐶 ) ) |
9 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ) → 𝜑 ) |
10 |
|
equid |
⊢ 𝑔 = 𝑔 |
11 |
|
rnuni |
⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 |
12 |
11
|
oveq1i |
⊢ ( ran ∪ 𝑋 ↑m 𝐶 ) = ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) |
13 |
10 12
|
eleq12i |
⊢ ( 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ↔ 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ) |
14 |
13
|
biimpi |
⊢ ( 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) → 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ) → 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ) |
16 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐵 ↑m 𝐶 ) ∈ V ) |
17 |
16 4
|
ssexd |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
18 |
|
rnexg |
⊢ ( 𝑓 ∈ 𝑋 → ran 𝑓 ∈ V ) |
19 |
18
|
rgen |
⊢ ∀ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V |
20 |
19
|
a1i |
⊢ ( 𝜑 → ∀ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V ) |
21 |
|
iunexg |
⊢ ( ( 𝑋 ∈ V ∧ ∀ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V ) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V ) |
22 |
17 20 21
|
syl2anc |
⊢ ( 𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V ) |
23 |
22 7
|
elmapd |
⊢ ( 𝜑 → ( 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ↔ 𝑔 : 𝐶 ⟶ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ) ) |
24 |
23
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ) → 𝑔 : 𝐶 ⟶ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ) |
25 |
|
snidg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 } ) |
26 |
1 25
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ { 𝐴 } ) |
27 |
26 3
|
eleqtrrdi |
⊢ ( 𝜑 → 𝐴 ∈ 𝐶 ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ) → 𝐴 ∈ 𝐶 ) |
29 |
24 28
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ) → ( 𝑔 ‘ 𝐴 ) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ) |
30 |
|
eliun |
⊢ ( ( 𝑔 ‘ 𝐴 ) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃ 𝑓 ∈ 𝑋 ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) |
31 |
29 30
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶 ) ) → ∃ 𝑓 ∈ 𝑋 ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) |
32 |
9 15 31
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ) → ∃ 𝑓 ∈ 𝑋 ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) |
33 |
|
elmapfn |
⊢ ( 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) → 𝑔 Fn 𝐶 ) |
34 |
33
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ) → 𝑔 Fn 𝐶 ) |
35 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) |
36 |
1
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝐴 ∈ 𝑉 ) |
37 |
3
|
oveq2i |
⊢ ( 𝐵 ↑m 𝐶 ) = ( 𝐵 ↑m { 𝐴 } ) |
38 |
4 37
|
sseqtrdi |
⊢ ( 𝜑 → 𝑋 ⊆ ( 𝐵 ↑m { 𝐴 } ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → 𝑋 ⊆ ( 𝐵 ↑m { 𝐴 } ) ) |
40 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → 𝑓 ∈ 𝑋 ) |
41 |
39 40
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → 𝑓 ∈ ( 𝐵 ↑m { 𝐴 } ) ) |
42 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → 𝐵 ∈ 𝑊 ) |
43 |
5
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → { 𝐴 } ∈ V ) |
44 |
42 43
|
elmapd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → ( 𝑓 ∈ ( 𝐵 ↑m { 𝐴 } ) ↔ 𝑓 : { 𝐴 } ⟶ 𝐵 ) ) |
45 |
41 44
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → 𝑓 : { 𝐴 } ⟶ 𝐵 ) |
46 |
45
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝑓 : { 𝐴 } ⟶ 𝐵 ) |
47 |
36 46
|
rnsnf |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → ran 𝑓 = { ( 𝑓 ‘ 𝐴 ) } ) |
48 |
35 47
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → ( 𝑔 ‘ 𝐴 ) ∈ { ( 𝑓 ‘ 𝐴 ) } ) |
49 |
|
fvex |
⊢ ( 𝑔 ‘ 𝐴 ) ∈ V |
50 |
49
|
elsn |
⊢ ( ( 𝑔 ‘ 𝐴 ) ∈ { ( 𝑓 ‘ 𝐴 ) } ↔ ( 𝑔 ‘ 𝐴 ) = ( 𝑓 ‘ 𝐴 ) ) |
51 |
48 50
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → ( 𝑔 ‘ 𝐴 ) = ( 𝑓 ‘ 𝐴 ) ) |
52 |
51
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → ( 𝑔 ‘ 𝐴 ) = ( 𝑓 ‘ 𝐴 ) ) |
53 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) → 𝐴 ∈ 𝑉 ) |
54 |
53
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝐴 ∈ 𝑉 ) |
55 |
|
simp1r |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝑔 Fn 𝐶 ) |
56 |
41 37
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → 𝑓 ∈ ( 𝐵 ↑m 𝐶 ) ) |
57 |
|
elmapfn |
⊢ ( 𝑓 ∈ ( 𝐵 ↑m 𝐶 ) → 𝑓 Fn 𝐶 ) |
58 |
56 57
|
syl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑋 ) → 𝑓 Fn 𝐶 ) |
59 |
58
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ) → 𝑓 Fn 𝐶 ) |
60 |
59
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝑓 Fn 𝐶 ) |
61 |
54 3 55 60
|
fsneq |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → ( 𝑔 = 𝑓 ↔ ( 𝑔 ‘ 𝐴 ) = ( 𝑓 ‘ 𝐴 ) ) ) |
62 |
52 61
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝑔 = 𝑓 ) |
63 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝑓 ∈ 𝑋 ) |
64 |
62 63
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) ∧ 𝑓 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 ) → 𝑔 ∈ 𝑋 ) |
65 |
64
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑔 Fn 𝐶 ) → ( 𝑓 ∈ 𝑋 → ( ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 → 𝑔 ∈ 𝑋 ) ) ) |
66 |
9 34 65
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ) → ( 𝑓 ∈ 𝑋 → ( ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 → 𝑔 ∈ 𝑋 ) ) ) |
67 |
66
|
rexlimdv |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ) → ( ∃ 𝑓 ∈ 𝑋 ( 𝑔 ‘ 𝐴 ) ∈ ran 𝑓 → 𝑔 ∈ 𝑋 ) ) |
68 |
32 67
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐶 ) ) → 𝑔 ∈ 𝑋 ) |
69 |
8 68
|
eqelssd |
⊢ ( 𝜑 → 𝑋 = ( ran ∪ 𝑋 ↑m 𝐶 ) ) |