Metamath Proof Explorer


Theorem cvmlift3lem4

Description: Lemma for cvmlift2 . (Contributed by Mario Carneiro, 6-Jul-2015)

Ref Expression
Hypotheses cvmlift3.b 𝐵 = 𝐶
cvmlift3.y 𝑌 = 𝐾
cvmlift3.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
cvmlift3.k ( 𝜑𝐾 ∈ SConn )
cvmlift3.l ( 𝜑𝐾 ∈ 𝑛-Locally PConn )
cvmlift3.o ( 𝜑𝑂𝑌 )
cvmlift3.g ( 𝜑𝐺 ∈ ( 𝐾 Cn 𝐽 ) )
cvmlift3.p ( 𝜑𝑃𝐵 )
cvmlift3.e ( 𝜑 → ( 𝐹𝑃 ) = ( 𝐺𝑂 ) )
cvmlift3.h 𝐻 = ( 𝑥𝑌 ↦ ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑥 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) )
Assertion cvmlift3lem4 ( ( 𝜑𝑋𝑌 ) → ( ( 𝐻𝑋 ) = 𝐴 ↔ ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 cvmlift3.b 𝐵 = 𝐶
2 cvmlift3.y 𝑌 = 𝐾
3 cvmlift3.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
4 cvmlift3.k ( 𝜑𝐾 ∈ SConn )
5 cvmlift3.l ( 𝜑𝐾 ∈ 𝑛-Locally PConn )
6 cvmlift3.o ( 𝜑𝑂𝑌 )
7 cvmlift3.g ( 𝜑𝐺 ∈ ( 𝐾 Cn 𝐽 ) )
8 cvmlift3.p ( 𝜑𝑃𝐵 )
9 cvmlift3.e ( 𝜑 → ( 𝐹𝑃 ) = ( 𝐺𝑂 ) )
10 cvmlift3.h 𝐻 = ( 𝑥𝑌 ↦ ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑥 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) )
11 1 2 3 4 5 6 7 8 9 10 cvmlift3lem3 ( 𝜑𝐻 : 𝑌𝐵 )
12 11 ffvelrnda ( ( 𝜑𝑋𝑌 ) → ( 𝐻𝑋 ) ∈ 𝐵 )
13 eleq1 ( ( 𝐻𝑋 ) = 𝐴 → ( ( 𝐻𝑋 ) ∈ 𝐵𝐴𝐵 ) )
14 12 13 syl5ibcom ( ( 𝜑𝑋𝑌 ) → ( ( 𝐻𝑋 ) = 𝐴𝐴𝐵 ) )
15 eqid ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) = ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) )
16 3 ad2antrr ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
17 simprl ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → 𝑓 ∈ ( II Cn 𝐾 ) )
18 7 ad2antrr ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → 𝐺 ∈ ( 𝐾 Cn 𝐽 ) )
19 cnco ( ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ 𝐺 ∈ ( 𝐾 Cn 𝐽 ) ) → ( 𝐺𝑓 ) ∈ ( II Cn 𝐽 ) )
20 17 18 19 syl2anc ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( 𝐺𝑓 ) ∈ ( II Cn 𝐽 ) )
21 8 ad2antrr ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → 𝑃𝐵 )
22 simprr ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( 𝑓 ‘ 0 ) = 𝑂 )
23 22 fveq2d ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( 𝐺 ‘ ( 𝑓 ‘ 0 ) ) = ( 𝐺𝑂 ) )
24 iiuni ( 0 [,] 1 ) = II
25 24 2 cnf ( 𝑓 ∈ ( II Cn 𝐾 ) → 𝑓 : ( 0 [,] 1 ) ⟶ 𝑌 )
26 17 25 syl ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → 𝑓 : ( 0 [,] 1 ) ⟶ 𝑌 )
27 0elunit 0 ∈ ( 0 [,] 1 )
28 fvco3 ( ( 𝑓 : ( 0 [,] 1 ) ⟶ 𝑌 ∧ 0 ∈ ( 0 [,] 1 ) ) → ( ( 𝐺𝑓 ) ‘ 0 ) = ( 𝐺 ‘ ( 𝑓 ‘ 0 ) ) )
29 26 27 28 sylancl ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( ( 𝐺𝑓 ) ‘ 0 ) = ( 𝐺 ‘ ( 𝑓 ‘ 0 ) ) )
30 9 ad2antrr ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( 𝐹𝑃 ) = ( 𝐺𝑂 ) )
31 23 29 30 3eqtr4rd ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( 𝐹𝑃 ) = ( ( 𝐺𝑓 ) ‘ 0 ) )
32 1 15 16 20 21 31 cvmliftiota ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ∈ ( II Cn 𝐶 ) ∧ ( 𝐹 ∘ ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ) = ( 𝐺𝑓 ) ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 0 ) = 𝑃 ) )
33 32 simp1d ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ∈ ( II Cn 𝐶 ) )
34 24 1 cnf ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ∈ ( II Cn 𝐶 ) → ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) : ( 0 [,] 1 ) ⟶ 𝐵 )
35 33 34 syl ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) : ( 0 [,] 1 ) ⟶ 𝐵 )
36 1elunit 1 ∈ ( 0 [,] 1 )
37 ffvelrn ( ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) : ( 0 [,] 1 ) ⟶ 𝐵 ∧ 1 ∈ ( 0 [,] 1 ) ) → ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) ∈ 𝐵 )
38 35 36 37 sylancl ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) ∈ 𝐵 )
39 eleq1 ( ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 → ( ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) ∈ 𝐵𝐴𝐵 ) )
40 38 39 syl5ibcom ( ( ( 𝜑𝑋𝑌 ) ∧ ( 𝑓 ∈ ( II Cn 𝐾 ) ∧ ( 𝑓 ‘ 0 ) = 𝑂 ) ) → ( ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴𝐴𝐵 ) )
41 40 expr ( ( ( 𝜑𝑋𝑌 ) ∧ 𝑓 ∈ ( II Cn 𝐾 ) ) → ( ( 𝑓 ‘ 0 ) = 𝑂 → ( ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴𝐴𝐵 ) ) )
42 41 a1dd ( ( ( 𝜑𝑋𝑌 ) ∧ 𝑓 ∈ ( II Cn 𝐾 ) ) → ( ( 𝑓 ‘ 0 ) = 𝑂 → ( ( 𝑓 ‘ 1 ) = 𝑋 → ( ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴𝐴𝐵 ) ) ) )
43 42 3impd ( ( ( 𝜑𝑋𝑌 ) ∧ 𝑓 ∈ ( II Cn 𝐾 ) ) → ( ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) → 𝐴𝐵 ) )
44 43 rexlimdva ( ( 𝜑𝑋𝑌 ) → ( ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) → 𝐴𝐵 ) )
45 eqeq2 ( 𝑥 = 𝑋 → ( ( 𝑓 ‘ 1 ) = 𝑥 ↔ ( 𝑓 ‘ 1 ) = 𝑋 ) )
46 45 3anbi2d ( 𝑥 = 𝑋 → ( ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑥 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ↔ ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) )
47 46 rexbidv ( 𝑥 = 𝑋 → ( ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑥 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ↔ ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) )
48 47 riotabidv ( 𝑥 = 𝑋 → ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑥 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) = ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) )
49 riotaex ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) ∈ V
50 48 10 49 fvmpt ( 𝑋𝑌 → ( 𝐻𝑋 ) = ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) )
51 50 adantl ( ( 𝜑𝑋𝑌 ) → ( 𝐻𝑋 ) = ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) )
52 51 eqeq1d ( ( 𝜑𝑋𝑌 ) → ( ( 𝐻𝑋 ) = 𝐴 ↔ ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) = 𝐴 ) )
53 52 adantl ( ( 𝐴𝐵 ∧ ( 𝜑𝑋𝑌 ) ) → ( ( 𝐻𝑋 ) = 𝐴 ↔ ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) = 𝐴 ) )
54 1 2 3 4 5 6 7 8 9 cvmlift3lem2 ( ( 𝜑𝑋𝑌 ) → ∃! 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) )
55 eqeq2 ( 𝑧 = 𝐴 → ( ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ↔ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) )
56 55 3anbi3d ( 𝑧 = 𝐴 → ( ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ↔ ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ) )
57 56 rexbidv ( 𝑧 = 𝐴 → ( ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ↔ ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ) )
58 57 riota2 ( ( 𝐴𝐵 ∧ ∃! 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) → ( ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ↔ ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) = 𝐴 ) )
59 54 58 sylan2 ( ( 𝐴𝐵 ∧ ( 𝜑𝑋𝑌 ) ) → ( ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ↔ ( 𝑧𝐵𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝑧 ) ) = 𝐴 ) )
60 53 59 bitr4d ( ( 𝐴𝐵 ∧ ( 𝜑𝑋𝑌 ) ) → ( ( 𝐻𝑋 ) = 𝐴 ↔ ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ) )
61 60 expcom ( ( 𝜑𝑋𝑌 ) → ( 𝐴𝐵 → ( ( 𝐻𝑋 ) = 𝐴 ↔ ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ) ) )
62 14 44 61 pm5.21ndd ( ( 𝜑𝑋𝑌 ) → ( ( 𝐻𝑋 ) = 𝐴 ↔ ∃ 𝑓 ∈ ( II Cn 𝐾 ) ( ( 𝑓 ‘ 0 ) = 𝑂 ∧ ( 𝑓 ‘ 1 ) = 𝑋 ∧ ( ( 𝑔 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑔 ) = ( 𝐺𝑓 ) ∧ ( 𝑔 ‘ 0 ) = 𝑃 ) ) ‘ 1 ) = 𝐴 ) ) )