Step |
Hyp |
Ref |
Expression |
1 |
|
unirnmap.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
unirnmap.x |
⊢ ( 𝜑 → 𝑋 ⊆ ( 𝐵 ↑m 𝐴 ) ) |
3 |
2
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → 𝑔 ∈ ( 𝐵 ↑m 𝐴 ) ) |
4 |
|
elmapfn |
⊢ ( 𝑔 ∈ ( 𝐵 ↑m 𝐴 ) → 𝑔 Fn 𝐴 ) |
5 |
3 4
|
syl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → 𝑔 Fn 𝐴 ) |
6 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑔 ∈ 𝑋 ) |
7 |
|
dffn3 |
⊢ ( 𝑔 Fn 𝐴 ↔ 𝑔 : 𝐴 ⟶ ran 𝑔 ) |
8 |
5 7
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → 𝑔 : 𝐴 ⟶ ran 𝑔 ) |
9 |
8
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑔 ‘ 𝑥 ) ∈ ran 𝑔 ) |
10 |
|
rneq |
⊢ ( 𝑓 = 𝑔 → ran 𝑓 = ran 𝑔 ) |
11 |
10
|
eleq2d |
⊢ ( 𝑓 = 𝑔 → ( ( 𝑔 ‘ 𝑥 ) ∈ ran 𝑓 ↔ ( 𝑔 ‘ 𝑥 ) ∈ ran 𝑔 ) ) |
12 |
11
|
rspcev |
⊢ ( ( 𝑔 ∈ 𝑋 ∧ ( 𝑔 ‘ 𝑥 ) ∈ ran 𝑔 ) → ∃ 𝑓 ∈ 𝑋 ( 𝑔 ‘ 𝑥 ) ∈ ran 𝑓 ) |
13 |
6 9 12
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑓 ∈ 𝑋 ( 𝑔 ‘ 𝑥 ) ∈ ran 𝑓 ) |
14 |
|
eliun |
⊢ ( ( 𝑔 ‘ 𝑥 ) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃ 𝑓 ∈ 𝑋 ( 𝑔 ‘ 𝑥 ) ∈ ran 𝑓 ) |
15 |
13 14
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑔 ‘ 𝑥 ) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ) |
16 |
|
rnuni |
⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 |
17 |
15 16
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑔 ‘ 𝑥 ) ∈ ran ∪ 𝑋 ) |
18 |
17
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ran ∪ 𝑋 ) |
19 |
5 18
|
jca |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → ( 𝑔 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ran ∪ 𝑋 ) ) |
20 |
|
ffnfv |
⊢ ( 𝑔 : 𝐴 ⟶ ran ∪ 𝑋 ↔ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ran ∪ 𝑋 ) ) |
21 |
19 20
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → 𝑔 : 𝐴 ⟶ ran ∪ 𝑋 ) |
22 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐵 ↑m 𝐴 ) ∈ V ) |
23 |
22 2
|
ssexd |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
24 |
23
|
uniexd |
⊢ ( 𝜑 → ∪ 𝑋 ∈ V ) |
25 |
|
rnexg |
⊢ ( ∪ 𝑋 ∈ V → ran ∪ 𝑋 ∈ V ) |
26 |
24 25
|
syl |
⊢ ( 𝜑 → ran ∪ 𝑋 ∈ V ) |
27 |
26 1
|
elmapd |
⊢ ( 𝜑 → ( 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐴 ) ↔ 𝑔 : 𝐴 ⟶ ran ∪ 𝑋 ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → ( 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐴 ) ↔ 𝑔 : 𝐴 ⟶ ran ∪ 𝑋 ) ) |
29 |
21 28
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝑋 ) → 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐴 ) ) |
30 |
29
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑔 ∈ 𝑋 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐴 ) ) |
31 |
|
dfss3 |
⊢ ( 𝑋 ⊆ ( ran ∪ 𝑋 ↑m 𝐴 ) ↔ ∀ 𝑔 ∈ 𝑋 𝑔 ∈ ( ran ∪ 𝑋 ↑m 𝐴 ) ) |
32 |
30 31
|
sylibr |
⊢ ( 𝜑 → 𝑋 ⊆ ( ran ∪ 𝑋 ↑m 𝐴 ) ) |