Step |
Hyp |
Ref |
Expression |
1 |
|
fsuppcurry1.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ ( 𝐶 𝐹 𝑥 ) ) |
2 |
|
fsuppcurry1.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑈 ) |
3 |
|
fsuppcurry1.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
fsuppcurry1.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
5 |
|
fsuppcurry1.f |
⊢ ( 𝜑 → 𝐹 Fn ( 𝐴 × 𝐵 ) ) |
6 |
|
fsuppcurry1.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
7 |
|
fsuppcurry1.1 |
⊢ ( 𝜑 → 𝐹 finSupp 𝑍 ) |
8 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐶 𝐹 𝑥 ) = ( 𝐶 𝐹 𝑦 ) ) |
9 |
8
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝐵 ↦ ( 𝐶 𝐹 𝑥 ) ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝐶 𝐹 𝑦 ) ) |
10 |
1 9
|
eqtri |
⊢ 𝐺 = ( 𝑦 ∈ 𝐵 ↦ ( 𝐶 𝐹 𝑦 ) ) |
11 |
4
|
mptexd |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↦ ( 𝐶 𝐹 𝑦 ) ) ∈ V ) |
12 |
10 11
|
eqeltrid |
⊢ ( 𝜑 → 𝐺 ∈ V ) |
13 |
1
|
funmpt2 |
⊢ Fun 𝐺 |
14 |
13
|
a1i |
⊢ ( 𝜑 → Fun 𝐺 ) |
15 |
|
fo2nd |
⊢ 2nd : V –onto→ V |
16 |
|
fofun |
⊢ ( 2nd : V –onto→ V → Fun 2nd ) |
17 |
15 16
|
ax-mp |
⊢ Fun 2nd |
18 |
|
funres |
⊢ ( Fun 2nd → Fun ( 2nd ↾ ( V × V ) ) ) |
19 |
17 18
|
mp1i |
⊢ ( 𝜑 → Fun ( 2nd ↾ ( V × V ) ) ) |
20 |
7
|
fsuppimpd |
⊢ ( 𝜑 → ( 𝐹 supp 𝑍 ) ∈ Fin ) |
21 |
|
imafi |
⊢ ( ( Fun ( 2nd ↾ ( V × V ) ) ∧ ( 𝐹 supp 𝑍 ) ∈ Fin ) → ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ∈ Fin ) |
22 |
19 20 21
|
syl2anc |
⊢ ( 𝜑 → ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ∈ Fin ) |
23 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝐶 𝐹 𝑦 ) ∈ V ) |
24 |
23 10
|
fmptd |
⊢ ( 𝜑 → 𝐺 : 𝐵 ⟶ V ) |
25 |
|
eldif |
⊢ ( 𝑦 ∈ ( 𝐵 ∖ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) ↔ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) ) |
26 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → 𝐶 ∈ 𝐴 ) |
27 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → 𝑦 ∈ 𝐵 ) |
28 |
26 27
|
opelxpd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → 〈 𝐶 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) |
29 |
|
df-ov |
⊢ ( 𝐶 𝐹 𝑦 ) = ( 𝐹 ‘ 〈 𝐶 , 𝑦 〉 ) |
30 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ( 𝐶 𝐹 𝑦 ) ∈ V ) |
31 |
1 8 27 30
|
fvmptd3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐶 𝐹 𝑦 ) ) |
32 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) |
33 |
32
|
neqned |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ( 𝐺 ‘ 𝑦 ) ≠ 𝑍 ) |
34 |
31 33
|
eqnetrrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ( 𝐶 𝐹 𝑦 ) ≠ 𝑍 ) |
35 |
29 34
|
eqnetrrid |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ( 𝐹 ‘ 〈 𝐶 , 𝑦 〉 ) ≠ 𝑍 ) |
36 |
3 4
|
xpexd |
⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ V ) |
37 |
|
elsuppfn |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ ( 𝐴 × 𝐵 ) ∈ V ∧ 𝑍 ∈ 𝑈 ) → ( 〈 𝐶 , 𝑦 〉 ∈ ( 𝐹 supp 𝑍 ) ↔ ( 〈 𝐶 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ∧ ( 𝐹 ‘ 〈 𝐶 , 𝑦 〉 ) ≠ 𝑍 ) ) ) |
38 |
5 36 2 37
|
syl3anc |
⊢ ( 𝜑 → ( 〈 𝐶 , 𝑦 〉 ∈ ( 𝐹 supp 𝑍 ) ↔ ( 〈 𝐶 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ∧ ( 𝐹 ‘ 〈 𝐶 , 𝑦 〉 ) ≠ 𝑍 ) ) ) |
39 |
38
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ( 〈 𝐶 , 𝑦 〉 ∈ ( 𝐹 supp 𝑍 ) ↔ ( 〈 𝐶 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ∧ ( 𝐹 ‘ 〈 𝐶 , 𝑦 〉 ) ≠ 𝑍 ) ) ) |
40 |
28 35 39
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → 〈 𝐶 , 𝑦 〉 ∈ ( 𝐹 supp 𝑍 ) ) |
41 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → 𝑧 = 〈 𝐶 , 𝑦 〉 ) |
42 |
41
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → ( ( 2nd ↾ ( V × V ) ) ‘ 𝑧 ) = ( ( 2nd ↾ ( V × V ) ) ‘ 〈 𝐶 , 𝑦 〉 ) ) |
43 |
|
xpss |
⊢ ( 𝐴 × 𝐵 ) ⊆ ( V × V ) |
44 |
28
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → 〈 𝐶 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) |
45 |
43 44
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → 〈 𝐶 , 𝑦 〉 ∈ ( V × V ) ) |
46 |
45
|
fvresd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → ( ( 2nd ↾ ( V × V ) ) ‘ 〈 𝐶 , 𝑦 〉 ) = ( 2nd ‘ 〈 𝐶 , 𝑦 〉 ) ) |
47 |
26
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → 𝐶 ∈ 𝐴 ) |
48 |
27
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → 𝑦 ∈ 𝐵 ) |
49 |
|
op2ndg |
⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝐶 , 𝑦 〉 ) = 𝑦 ) |
50 |
47 48 49
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → ( 2nd ‘ 〈 𝐶 , 𝑦 〉 ) = 𝑦 ) |
51 |
42 46 50
|
3eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ∧ 𝑧 = 〈 𝐶 , 𝑦 〉 ) → ( ( 2nd ↾ ( V × V ) ) ‘ 𝑧 ) = 𝑦 ) |
52 |
40 51
|
rspcedeq1vd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ∃ 𝑧 ∈ ( 𝐹 supp 𝑍 ) ( ( 2nd ↾ ( V × V ) ) ‘ 𝑧 ) = 𝑦 ) |
53 |
|
fofn |
⊢ ( 2nd : V –onto→ V → 2nd Fn V ) |
54 |
|
fnresin |
⊢ ( 2nd Fn V → ( 2nd ↾ ( V × V ) ) Fn ( V ∩ ( V × V ) ) ) |
55 |
15 53 54
|
mp2b |
⊢ ( 2nd ↾ ( V × V ) ) Fn ( V ∩ ( V × V ) ) |
56 |
|
ssv |
⊢ ( V × V ) ⊆ V |
57 |
|
sseqin2 |
⊢ ( ( V × V ) ⊆ V ↔ ( V ∩ ( V × V ) ) = ( V × V ) ) |
58 |
56 57
|
mpbi |
⊢ ( V ∩ ( V × V ) ) = ( V × V ) |
59 |
58
|
fneq2i |
⊢ ( ( 2nd ↾ ( V × V ) ) Fn ( V ∩ ( V × V ) ) ↔ ( 2nd ↾ ( V × V ) ) Fn ( V × V ) ) |
60 |
55 59
|
mpbi |
⊢ ( 2nd ↾ ( V × V ) ) Fn ( V × V ) |
61 |
60
|
a1i |
⊢ ( 𝜑 → ( 2nd ↾ ( V × V ) ) Fn ( V × V ) ) |
62 |
|
suppssdm |
⊢ ( 𝐹 supp 𝑍 ) ⊆ dom 𝐹 |
63 |
5
|
fndmd |
⊢ ( 𝜑 → dom 𝐹 = ( 𝐴 × 𝐵 ) ) |
64 |
62 63
|
sseqtrid |
⊢ ( 𝜑 → ( 𝐹 supp 𝑍 ) ⊆ ( 𝐴 × 𝐵 ) ) |
65 |
64 43
|
sstrdi |
⊢ ( 𝜑 → ( 𝐹 supp 𝑍 ) ⊆ ( V × V ) ) |
66 |
61 65
|
fvelimabd |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ↔ ∃ 𝑧 ∈ ( 𝐹 supp 𝑍 ) ( ( 2nd ↾ ( V × V ) ) ‘ 𝑧 ) = 𝑦 ) ) |
67 |
66
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → ( 𝑦 ∈ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ↔ ∃ 𝑧 ∈ ( 𝐹 supp 𝑍 ) ( ( 2nd ↾ ( V × V ) ) ‘ 𝑧 ) = 𝑦 ) ) |
68 |
52 67
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 ) → 𝑦 ∈ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) |
69 |
68
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ¬ ( 𝐺 ‘ 𝑦 ) = 𝑍 → 𝑦 ∈ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) ) |
70 |
69
|
con1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ¬ 𝑦 ∈ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) → ( 𝐺 ‘ 𝑦 ) = 𝑍 ) ) |
71 |
70
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) ) → ( 𝐺 ‘ 𝑦 ) = 𝑍 ) |
72 |
25 71
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 ∖ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) ) → ( 𝐺 ‘ 𝑦 ) = 𝑍 ) |
73 |
24 72
|
suppss |
⊢ ( 𝜑 → ( 𝐺 supp 𝑍 ) ⊆ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) |
74 |
|
suppssfifsupp |
⊢ ( ( ( 𝐺 ∈ V ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑈 ) ∧ ( ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ∈ Fin ∧ ( 𝐺 supp 𝑍 ) ⊆ ( ( 2nd ↾ ( V × V ) ) “ ( 𝐹 supp 𝑍 ) ) ) ) → 𝐺 finSupp 𝑍 ) |
75 |
12 14 2 22 73 74
|
syl32anc |
⊢ ( 𝜑 → 𝐺 finSupp 𝑍 ) |