| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | dfmgc2lem.1 | . . 3
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| 2 |  | dfmgc2lem.2 | . . 3
⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | 
| 3 | 1, 2 | jca 511 | . 2
⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) | 
| 4 |  | mgcval.2 | . . . . . . 7
⊢ (𝜑 → 𝑉 ∈ Proset ) | 
| 5 | 4 | ad3antrrr 730 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → 𝑉 ∈ Proset ) | 
| 6 |  | simplr 769 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑧 ∈ 𝐴) | 
| 7 | 6 | adantr 480 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → 𝑧 ∈ 𝐴) | 
| 8 | 2 | ad3antrrr 730 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → 𝐺:𝐵⟶𝐴) | 
| 9 | 1 | ad3antrrr 730 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → 𝐹:𝐴⟶𝐵) | 
| 10 | 9, 7 | ffvelcdmd 7105 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → (𝐹‘𝑧) ∈ 𝐵) | 
| 11 | 8, 10 | ffvelcdmd 7105 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → (𝐺‘(𝐹‘𝑧)) ∈ 𝐴) | 
| 12 | 2 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝐺:𝐵⟶𝐴) | 
| 13 |  | simpr 484 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵) | 
| 14 | 12, 13 | ffvelcdmd 7105 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝐺‘𝑤) ∈ 𝐴) | 
| 15 | 14 | adantr 480 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → (𝐺‘𝑤) ∈ 𝐴) | 
| 16 |  | dfmgc2lem.5 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) | 
| 17 | 16 | ralrimiva 3146 | . . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) | 
| 18 | 17 | ad3antrrr 730 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) | 
| 19 |  | simpr 484 | . . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧) | 
| 20 | 19 | fveq2d 6910 | . . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) ∧ 𝑥 = 𝑧) → (𝐹‘𝑥) = (𝐹‘𝑧)) | 
| 21 | 20 | fveq2d 6910 | . . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) ∧ 𝑥 = 𝑧) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑧))) | 
| 22 | 19, 21 | breq12d 5156 | . . . . . . . 8
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) ∧ 𝑥 = 𝑧) → (𝑥 ≤ (𝐺‘(𝐹‘𝑥)) ↔ 𝑧 ≤ (𝐺‘(𝐹‘𝑧)))) | 
| 23 | 7, 22 | rspcdv 3614 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → (∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)) → 𝑧 ≤ (𝐺‘(𝐹‘𝑧)))) | 
| 24 | 18, 23 | mpd 15 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → 𝑧 ≤ (𝐺‘(𝐹‘𝑧))) | 
| 25 |  | dfmgc2lem.4 | . . . . . . . . 9
⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) | 
| 26 | 25 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) | 
| 27 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑢 = (𝐹‘𝑧) → (𝑢 ≲ 𝑣 ↔ (𝐹‘𝑧) ≲ 𝑣)) | 
| 28 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑢 = (𝐹‘𝑧) → (𝐺‘𝑢) = (𝐺‘(𝐹‘𝑧))) | 
| 29 | 28 | breq1d 5153 | . . . . . . . . . 10
⊢ (𝑢 = (𝐹‘𝑧) → ((𝐺‘𝑢) ≤ (𝐺‘𝑣) ↔ (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑣))) | 
| 30 | 27, 29 | imbi12d 344 | . . . . . . . . 9
⊢ (𝑢 = (𝐹‘𝑧) → ((𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)) ↔ ((𝐹‘𝑧) ≲ 𝑣 → (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑣)))) | 
| 31 |  | breq2 5147 | . . . . . . . . . 10
⊢ (𝑣 = 𝑤 → ((𝐹‘𝑧) ≲ 𝑣 ↔ (𝐹‘𝑧) ≲ 𝑤)) | 
| 32 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑣 = 𝑤 → (𝐺‘𝑣) = (𝐺‘𝑤)) | 
| 33 | 32 | breq2d 5155 | . . . . . . . . . 10
⊢ (𝑣 = 𝑤 → ((𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑣) ↔ (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑤))) | 
| 34 | 31, 33 | imbi12d 344 | . . . . . . . . 9
⊢ (𝑣 = 𝑤 → (((𝐹‘𝑧) ≲ 𝑣 → (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑣)) ↔ ((𝐹‘𝑧) ≲ 𝑤 → (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑤)))) | 
| 35 | 1 | ffvelcdmda 7104 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) | 
| 36 | 35 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝐹‘𝑧) ∈ 𝐵) | 
| 37 |  | eqidd 2738 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = (𝐹‘𝑧)) → 𝐵 = 𝐵) | 
| 38 | 30, 34, 36, 37, 13 | rspc2vd 3947 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)) → ((𝐹‘𝑧) ≲ 𝑤 → (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑤)))) | 
| 39 | 26, 38 | mpd 15 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) ≲ 𝑤 → (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑤))) | 
| 40 | 39 | imp 406 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑤)) | 
| 41 |  | mgcoval.1 | . . . . . . 7
⊢ 𝐴 = (Base‘𝑉) | 
| 42 |  | mgcoval.3 | . . . . . . 7
⊢  ≤ =
(le‘𝑉) | 
| 43 | 41, 42 | prstr 18345 | . . . . . 6
⊢ ((𝑉 ∈ Proset ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘(𝐹‘𝑧)) ∈ 𝐴 ∧ (𝐺‘𝑤) ∈ 𝐴) ∧ (𝑧 ≤ (𝐺‘(𝐹‘𝑧)) ∧ (𝐺‘(𝐹‘𝑧)) ≤ (𝐺‘𝑤))) → 𝑧 ≤ (𝐺‘𝑤)) | 
| 44 | 5, 7, 11, 15, 24, 40, 43 | syl132anc 1390 | . . . . 5
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ (𝐹‘𝑧) ≲ 𝑤) → 𝑧 ≤ (𝐺‘𝑤)) | 
| 45 |  | mgcval.3 | . . . . . . 7
⊢ (𝜑 → 𝑊 ∈ Proset ) | 
| 46 | 45 | ad3antrrr 730 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → 𝑊 ∈ Proset ) | 
| 47 | 35 | ad2antrr 726 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → (𝐹‘𝑧) ∈ 𝐵) | 
| 48 | 1 | ad3antrrr 730 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → 𝐹:𝐴⟶𝐵) | 
| 49 | 14 | adantr 480 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → (𝐺‘𝑤) ∈ 𝐴) | 
| 50 | 48, 49 | ffvelcdmd 7105 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → (𝐹‘(𝐺‘𝑤)) ∈ 𝐵) | 
| 51 |  | simplr 769 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → 𝑤 ∈ 𝐵) | 
| 52 |  | dfmgc2lem.3 | . . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))) | 
| 53 | 52 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))) | 
| 54 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦)) | 
| 55 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | 
| 56 | 55 | breq1d 5153 | . . . . . . . . . 10
⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ≲ (𝐹‘𝑦))) | 
| 57 | 54, 56 | imbi12d 344 | . . . . . . . . 9
⊢ (𝑥 = 𝑧 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑧 ≤ 𝑦 → (𝐹‘𝑧) ≲ (𝐹‘𝑦)))) | 
| 58 |  | breq2 5147 | . . . . . . . . . 10
⊢ (𝑦 = (𝐺‘𝑤) → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ (𝐺‘𝑤))) | 
| 59 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑦 = (𝐺‘𝑤) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑤))) | 
| 60 | 59 | breq2d 5155 | . . . . . . . . . 10
⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑧) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ≲ (𝐹‘(𝐺‘𝑤)))) | 
| 61 | 58, 60 | imbi12d 344 | . . . . . . . . 9
⊢ (𝑦 = (𝐺‘𝑤) → ((𝑧 ≤ 𝑦 → (𝐹‘𝑧) ≲ (𝐹‘𝑦)) ↔ (𝑧 ≤ (𝐺‘𝑤) → (𝐹‘𝑧) ≲ (𝐹‘(𝐺‘𝑤))))) | 
| 62 |  | eqidd 2738 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑥 = 𝑧) → 𝐴 = 𝐴) | 
| 63 | 57, 61, 6, 62, 14 | rspc2vd 3947 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) → (𝑧 ≤ (𝐺‘𝑤) → (𝐹‘𝑧) ≲ (𝐹‘(𝐺‘𝑤))))) | 
| 64 | 53, 63 | mpd 15 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝑧 ≤ (𝐺‘𝑤) → (𝐹‘𝑧) ≲ (𝐹‘(𝐺‘𝑤)))) | 
| 65 | 64 | imp 406 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → (𝐹‘𝑧) ≲ (𝐹‘(𝐺‘𝑤))) | 
| 66 |  | dfmgc2lem.6 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) | 
| 67 | 66 | ralrimiva 3146 | . . . . . . . 8
⊢ (𝜑 → ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) | 
| 68 | 67 | ad3antrrr 730 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) | 
| 69 |  | simpr 484 | . . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤) | 
| 70 | 69 | fveq2d 6910 | . . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) ∧ 𝑢 = 𝑤) → (𝐺‘𝑢) = (𝐺‘𝑤)) | 
| 71 | 70 | fveq2d 6910 | . . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) ∧ 𝑢 = 𝑤) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝑤))) | 
| 72 | 71, 69 | breq12d 5156 | . . . . . . . 8
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) ∧ 𝑢 = 𝑤) → ((𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ↔ (𝐹‘(𝐺‘𝑤)) ≲ 𝑤)) | 
| 73 | 51, 72 | rspcdv 3614 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 → (𝐹‘(𝐺‘𝑤)) ≲ 𝑤)) | 
| 74 | 68, 73 | mpd 15 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → (𝐹‘(𝐺‘𝑤)) ≲ 𝑤) | 
| 75 |  | mgcoval.2 | . . . . . . 7
⊢ 𝐵 = (Base‘𝑊) | 
| 76 |  | mgcoval.4 | . . . . . . 7
⊢  ≲ =
(le‘𝑊) | 
| 77 | 75, 76 | prstr 18345 | . . . . . 6
⊢ ((𝑊 ∈ Proset ∧ ((𝐹‘𝑧) ∈ 𝐵 ∧ (𝐹‘(𝐺‘𝑤)) ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ ((𝐹‘𝑧) ≲ (𝐹‘(𝐺‘𝑤)) ∧ (𝐹‘(𝐺‘𝑤)) ≲ 𝑤)) → (𝐹‘𝑧) ≲ 𝑤) | 
| 78 | 46, 47, 50, 51, 65, 74, 77 | syl132anc 1390 | . . . . 5
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) ∧ 𝑧 ≤ (𝐺‘𝑤)) → (𝐹‘𝑧) ≲ 𝑤) | 
| 79 | 44, 78 | impbida 801 | . . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) ≲ 𝑤 ↔ 𝑧 ≤ (𝐺‘𝑤))) | 
| 80 | 79 | anasss 466 | . . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → ((𝐹‘𝑧) ≲ 𝑤 ↔ 𝑧 ≤ (𝐺‘𝑤))) | 
| 81 | 80 | ralrimivva 3202 | . 2
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) ≲ 𝑤 ↔ 𝑧 ≤ (𝐺‘𝑤))) | 
| 82 |  | mgcval.1 | . . 3
⊢ 𝐻 = (𝑉MGalConn𝑊) | 
| 83 | 41, 75, 42, 76, 82, 4, 45 | mgcval 32977 | . 2
⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) ≲ 𝑤 ↔ 𝑧 ≤ (𝐺‘𝑤))))) | 
| 84 | 3, 81, 83 | mpbir2and 713 | 1
⊢ (𝜑 → 𝐹𝐻𝐺) |