Step |
Hyp |
Ref |
Expression |
1 |
|
fpwrelmapffslem.1 |
⊢ 𝐴 ∈ V |
2 |
|
fpwrelmapffslem.2 |
⊢ 𝐵 ∈ V |
3 |
|
fpwrelmapffslem.3 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝒫 𝐵 ) |
4 |
|
fpwrelmapffslem.4 |
⊢ ( 𝜑 → 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
5 |
|
relopab |
⊢ Rel { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
6 |
|
releq |
⊢ ( 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } → ( Rel 𝑅 ↔ Rel { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) ) |
7 |
5 6
|
mpbiri |
⊢ ( 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } → Rel 𝑅 ) |
8 |
|
relfi |
⊢ ( Rel 𝑅 → ( 𝑅 ∈ Fin ↔ ( dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin ) ) ) |
9 |
4 7 8
|
3syl |
⊢ ( 𝜑 → ( 𝑅 ∈ Fin ↔ ( dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin ) ) ) |
10 |
|
rexcom4 |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) |
11 |
|
ancom |
⊢ ( ( 𝑧 = ( 𝐹 ‘ 𝑥 ) ∧ 𝑤 ∈ 𝑧 ) ↔ ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) |
12 |
11
|
exbii |
⊢ ( ∃ 𝑧 ( 𝑧 = ( 𝐹 ‘ 𝑥 ) ∧ 𝑤 ∈ 𝑧 ) ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) |
13 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
14 |
|
eleq2 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑥 ) → ( 𝑤 ∈ 𝑧 ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
15 |
13 14
|
ceqsexv |
⊢ ( ∃ 𝑧 ( 𝑧 = ( 𝐹 ‘ 𝑥 ) ∧ 𝑤 ∈ 𝑧 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) |
16 |
12 15
|
bitr3i |
⊢ ( ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) |
17 |
16
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝐴 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) |
18 |
|
r19.42v |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) |
19 |
18
|
exbii |
⊢ ( ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) |
20 |
10 17 19
|
3bitr3ri |
⊢ ( ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝐴 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) |
21 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
22 |
20 21
|
bitr2i |
⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) |
23 |
22
|
a1i |
⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) ) |
24 |
|
vex |
⊢ 𝑤 ∈ V |
25 |
|
eleq1w |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
26 |
25
|
anbi2d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) |
27 |
26
|
exbidv |
⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) |
28 |
24 27
|
elab |
⊢ ( 𝑤 ∈ { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑤 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
29 |
|
eluniab |
⊢ ( 𝑤 ∈ ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) ) ) |
30 |
23 28 29
|
3bitr4g |
⊢ ( 𝜑 → ( 𝑤 ∈ { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↔ 𝑤 ∈ ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ) ) |
31 |
30
|
eqrdv |
⊢ ( 𝜑 → { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } = ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ) |
32 |
31
|
eleq1d |
⊢ ( 𝜑 → ( { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ∈ Fin ↔ ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ∈ Fin ↔ ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ) ) |
34 |
|
ffn |
⊢ ( 𝐹 : 𝐴 ⟶ 𝒫 𝐵 → 𝐹 Fn 𝐴 ) |
35 |
|
fnrnfv |
⊢ ( 𝐹 Fn 𝐴 → ran 𝐹 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ) |
36 |
3 34 35
|
3syl |
⊢ ( 𝜑 → ran 𝐹 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ran 𝐹 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ) |
38 |
|
0ex |
⊢ ∅ ∈ V |
39 |
38
|
a1i |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ∅ ∈ V ) |
40 |
|
fex |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝒫 𝐵 ∧ 𝐴 ∈ V ) → 𝐹 ∈ V ) |
41 |
3 1 40
|
sylancl |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → 𝐹 ∈ V ) |
43 |
3
|
ffund |
⊢ ( 𝜑 → Fun 𝐹 ) |
44 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → Fun 𝐹 ) |
45 |
|
opabdm |
⊢ ( 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } → dom 𝑅 = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
46 |
4 45
|
syl |
⊢ ( 𝜑 → dom 𝑅 = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
47 |
1 40
|
mpan2 |
⊢ ( 𝐹 : 𝐴 ⟶ 𝒫 𝐵 → 𝐹 ∈ V ) |
48 |
|
suppimacnv |
⊢ ( ( 𝐹 ∈ V ∧ ∅ ∈ V ) → ( 𝐹 supp ∅ ) = ( ◡ 𝐹 “ ( V ∖ { ∅ } ) ) ) |
49 |
38 48
|
mpan2 |
⊢ ( 𝐹 ∈ V → ( 𝐹 supp ∅ ) = ( ◡ 𝐹 “ ( V ∖ { ∅ } ) ) ) |
50 |
3 47 49
|
3syl |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) = ( ◡ 𝐹 “ ( V ∖ { ∅ } ) ) ) |
51 |
3
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
52 |
51
|
cnveqd |
⊢ ( 𝜑 → ◡ 𝐹 = ◡ ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
53 |
52
|
imaeq1d |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { ∅ } ) ) = ( ◡ ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) “ ( V ∖ { ∅ } ) ) ) |
54 |
50 53
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) = ( ◡ ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) “ ( V ∖ { ∅ } ) ) ) |
55 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) |
56 |
55
|
mptpreima |
⊢ ( ◡ ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) “ ( V ∖ { ∅ } ) ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ∈ ( V ∖ { ∅ } ) } |
57 |
54 56
|
eqtrdi |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ∈ ( V ∖ { ∅ } ) } ) |
58 |
|
suppvalfn |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ ∅ ∈ V ) → ( 𝐹 supp ∅ ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ∅ } ) |
59 |
1 38 58
|
mp3an23 |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 supp ∅ ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ∅ } ) |
60 |
3 34 59
|
3syl |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ∅ } ) |
61 |
|
n0 |
⊢ ( ( 𝐹 ‘ 𝑥 ) ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) |
62 |
61
|
rabbii |
⊢ { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ∅ } = { 𝑥 ∈ 𝐴 ∣ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } |
63 |
62
|
a1i |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ∅ } = { 𝑥 ∈ 𝐴 ∣ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) |
64 |
60 57 63
|
3eqtr3d |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ∈ ( V ∖ { ∅ } ) } = { 𝑥 ∈ 𝐴 ∣ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) |
65 |
|
df-rab |
⊢ { 𝑥 ∈ 𝐴 ∣ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
66 |
|
19.42v |
⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
67 |
66
|
abbii |
⊢ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
68 |
65 67
|
eqtr4i |
⊢ { 𝑥 ∈ 𝐴 ∣ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
69 |
68
|
a1i |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ∃ 𝑦 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
70 |
57 64 69
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
71 |
46 70
|
eqtr4d |
⊢ ( 𝜑 → dom 𝑅 = ( 𝐹 supp ∅ ) ) |
72 |
71
|
eleq1d |
⊢ ( 𝜑 → ( dom 𝑅 ∈ Fin ↔ ( 𝐹 supp ∅ ) ∈ Fin ) ) |
73 |
72
|
biimpa |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( 𝐹 supp ∅ ) ∈ Fin ) |
74 |
39 42 44 73
|
ffsrn |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ran 𝐹 ∈ Fin ) |
75 |
37 74
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ) |
76 |
|
unifi |
⊢ ( ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ∧ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ⊆ Fin ) → ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ) |
77 |
76
|
ex |
⊢ ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin → ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ⊆ Fin → ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ) ) |
78 |
75 77
|
syl |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ⊆ Fin → ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ) ) |
79 |
|
unifi3 |
⊢ ( ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ⊆ Fin ) |
80 |
78 79
|
impbid1 |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ⊆ Fin ↔ ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ∈ Fin ) ) |
81 |
33 80
|
bitr4d |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ∈ Fin ↔ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ⊆ Fin ) ) |
82 |
|
opabrn |
⊢ ( 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } → ran 𝑅 = { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
83 |
4 82
|
syl |
⊢ ( 𝜑 → ran 𝑅 = { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
84 |
83
|
eleq1d |
⊢ ( 𝜑 → ( ran 𝑅 ∈ Fin ↔ { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ∈ Fin ) ) |
85 |
84
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( ran 𝑅 ∈ Fin ↔ { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ∈ Fin ) ) |
86 |
37
|
sseq1d |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( ran 𝐹 ⊆ Fin ↔ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑥 ) } ⊆ Fin ) ) |
87 |
81 85 86
|
3bitr4d |
⊢ ( ( 𝜑 ∧ dom 𝑅 ∈ Fin ) → ( ran 𝑅 ∈ Fin ↔ ran 𝐹 ⊆ Fin ) ) |
88 |
87
|
pm5.32da |
⊢ ( 𝜑 → ( ( dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin ) ↔ ( dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin ) ) ) |
89 |
72
|
anbi1d |
⊢ ( 𝜑 → ( ( dom 𝑅 ∈ Fin ∧ ran 𝐹 ⊆ Fin ) ↔ ( ( 𝐹 supp ∅ ) ∈ Fin ∧ ran 𝐹 ⊆ Fin ) ) ) |
90 |
88 89
|
bitrd |
⊢ ( 𝜑 → ( ( dom 𝑅 ∈ Fin ∧ ran 𝑅 ∈ Fin ) ↔ ( ( 𝐹 supp ∅ ) ∈ Fin ∧ ran 𝐹 ⊆ Fin ) ) ) |
91 |
|
ancom |
⊢ ( ( ( 𝐹 supp ∅ ) ∈ Fin ∧ ran 𝐹 ⊆ Fin ) ↔ ( ran 𝐹 ⊆ Fin ∧ ( 𝐹 supp ∅ ) ∈ Fin ) ) |
92 |
91
|
a1i |
⊢ ( 𝜑 → ( ( ( 𝐹 supp ∅ ) ∈ Fin ∧ ran 𝐹 ⊆ Fin ) ↔ ( ran 𝐹 ⊆ Fin ∧ ( 𝐹 supp ∅ ) ∈ Fin ) ) ) |
93 |
9 90 92
|
3bitrd |
⊢ ( 𝜑 → ( 𝑅 ∈ Fin ↔ ( ran 𝐹 ⊆ Fin ∧ ( 𝐹 supp ∅ ) ∈ Fin ) ) ) |