Step |
Hyp |
Ref |
Expression |
1 |
|
sscfn1.1 |
⊢ ( 𝜑 → 𝐻 ⊆cat 𝐽 ) |
2 |
|
sscfn1.2 |
⊢ ( 𝜑 → 𝑆 = dom dom 𝐻 ) |
3 |
|
brssc |
⊢ ( 𝐻 ⊆cat 𝐽 ↔ ∃ 𝑡 ( 𝐽 Fn ( 𝑡 × 𝑡 ) ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑥 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑥 ) ) ) |
4 |
1 3
|
sylib |
⊢ ( 𝜑 → ∃ 𝑡 ( 𝐽 Fn ( 𝑡 × 𝑡 ) ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑥 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑥 ) ) ) |
5 |
|
ixpfn |
⊢ ( 𝐻 ∈ X 𝑥 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑥 ) → 𝐻 Fn ( 𝑠 × 𝑠 ) ) |
6 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → 𝐻 Fn ( 𝑠 × 𝑠 ) ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → 𝑆 = dom dom 𝐻 ) |
8 |
|
fndm |
⊢ ( 𝐻 Fn ( 𝑠 × 𝑠 ) → dom 𝐻 = ( 𝑠 × 𝑠 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → dom 𝐻 = ( 𝑠 × 𝑠 ) ) |
10 |
9
|
dmeqd |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → dom dom 𝐻 = dom ( 𝑠 × 𝑠 ) ) |
11 |
|
dmxpid |
⊢ dom ( 𝑠 × 𝑠 ) = 𝑠 |
12 |
10 11
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → dom dom 𝐻 = 𝑠 ) |
13 |
7 12
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → 𝑠 = 𝑆 ) |
14 |
13
|
sqxpeqd |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → ( 𝑠 × 𝑠 ) = ( 𝑆 × 𝑆 ) ) |
15 |
14
|
fneq2d |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → ( 𝐻 Fn ( 𝑠 × 𝑠 ) ↔ 𝐻 Fn ( 𝑆 × 𝑆 ) ) ) |
16 |
6 15
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |
17 |
16
|
ex |
⊢ ( 𝜑 → ( 𝐻 Fn ( 𝑠 × 𝑠 ) → 𝐻 Fn ( 𝑆 × 𝑆 ) ) ) |
18 |
5 17
|
syl5 |
⊢ ( 𝜑 → ( 𝐻 ∈ X 𝑥 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑥 ) → 𝐻 Fn ( 𝑆 × 𝑆 ) ) ) |
19 |
18
|
rexlimdvw |
⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑥 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑥 ) → 𝐻 Fn ( 𝑆 × 𝑆 ) ) ) |
20 |
19
|
adantld |
⊢ ( 𝜑 → ( ( 𝐽 Fn ( 𝑡 × 𝑡 ) ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑥 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑥 ) ) → 𝐻 Fn ( 𝑆 × 𝑆 ) ) ) |
21 |
20
|
exlimdv |
⊢ ( 𝜑 → ( ∃ 𝑡 ( 𝐽 Fn ( 𝑡 × 𝑡 ) ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑥 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑥 ) ) → 𝐻 Fn ( 𝑆 × 𝑆 ) ) ) |
22 |
4 21
|
mpd |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |