| Step | Hyp | Ref
| Expression |
| 1 | | simp1 1137 |
. . . 4
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–onto→𝐵) |
| 2 | | fof 6820 |
. . . 4
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) |
| 3 | 1, 2 | syl 17 |
. . 3
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴⟶𝐵) |
| 4 | | domnsym 9139 |
. . . . . . 7
⊢ (𝐵 ≼ (𝐴 ∖ {𝑦}) → ¬ (𝐴 ∖ {𝑦}) ≺ 𝐵) |
| 5 | | simp3 1139 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐵 ∈ Fin) |
| 6 | | simp2 1138 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵) |
| 7 | | enfii 9226 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) |
| 8 | 5, 6, 7 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin) |
| 9 | 8 | ad2antrr 726 |
. . . . . . . . 9
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ∈ Fin) |
| 10 | | difssd 4137 |
. . . . . . . . . 10
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ⊆ 𝐴) |
| 11 | | simplrr 778 |
. . . . . . . . . . . 12
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ 𝐴) |
| 12 | | neldifsn 4792 |
. . . . . . . . . . . 12
⊢ ¬
𝑦 ∈ (𝐴 ∖ {𝑦}) |
| 13 | | nelne1 3039 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})) → 𝐴 ≠ (𝐴 ∖ {𝑦})) |
| 14 | 11, 12, 13 | sylancl 586 |
. . . . . . . . . . 11
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ≠ (𝐴 ∖ {𝑦})) |
| 15 | 14 | necomd 2996 |
. . . . . . . . . 10
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≠ 𝐴) |
| 16 | | df-pss 3971 |
. . . . . . . . . 10
⊢ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ ((𝐴 ∖ {𝑦}) ⊆ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≠ 𝐴)) |
| 17 | 10, 15, 16 | sylanbrc 583 |
. . . . . . . . 9
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ⊊ 𝐴) |
| 18 | | php3 9249 |
. . . . . . . . 9
⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊊ 𝐴) → (𝐴 ∖ {𝑦}) ≺ 𝐴) |
| 19 | 9, 17, 18 | syl2anc 584 |
. . . . . . . 8
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐴) |
| 20 | 6 | ad2antrr 726 |
. . . . . . . 8
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐴 ≈ 𝐵) |
| 21 | | sdomentr 9151 |
. . . . . . . 8
⊢ (((𝐴 ∖ {𝑦}) ≺ 𝐴 ∧ 𝐴 ≈ 𝐵) → (𝐴 ∖ {𝑦}) ≺ 𝐵) |
| 22 | 19, 20, 21 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ {𝑦}) ≺ 𝐵) |
| 23 | 4, 22 | nsyl3 138 |
. . . . . 6
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 𝐵 ≼ (𝐴 ∖ {𝑦})) |
| 24 | 8 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐴 ∈ Fin) |
| 25 | | difss 4136 |
. . . . . . . . . . 11
⊢ (𝐴 ∖ {𝑦}) ⊆ 𝐴 |
| 26 | | ssfi 9213 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐴 ∖ {𝑦}) ∈ Fin) |
| 27 | 24, 25, 26 | sylancl 586 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐴 ∖ {𝑦}) ∈ Fin) |
| 28 | 3 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐹:𝐴⟶𝐵) |
| 29 | | fssres 6774 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∖ {𝑦}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵) |
| 30 | 28, 25, 29 | sylancl 586 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵) |
| 31 | 1 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐹:𝐴–onto→𝐵) |
| 32 | | foelrn 7127 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢)) |
| 33 | 31, 32 | sylan 580 |
. . . . . . . . . . . . 13
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢)) |
| 34 | | simprll 779 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ∈ 𝐴) |
| 35 | | simprrr 782 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ≠ 𝑦) |
| 36 | | eldifsn 4786 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝑦)) |
| 37 | 34, 35, 36 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝑥 ∈ (𝐴 ∖ {𝑦})) |
| 38 | | simprrl 781 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦)) |
| 39 | 38 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹‘𝑦) = (𝐹‘𝑥)) |
| 40 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥)) |
| 41 | 40 | rspceeqv 3645 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹‘𝑦) = (𝐹‘𝑥)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤)) |
| 42 | 37, 39, 41 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤)) |
| 43 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 = 𝑦 → ((𝐹‘𝑢) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑤))) |
| 44 | 43 | rexbidv 3179 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 𝑦 → (∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑦) = (𝐹‘𝑤))) |
| 45 | 42, 44 | syl5ibrcom 247 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))) |
| 46 | 45 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) → (𝑢 = 𝑦 → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))) |
| 47 | 46 | imp 406 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 = 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)) |
| 48 | | eldifsn 4786 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦)) |
| 49 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐹‘𝑢) = (𝐹‘𝑢) |
| 50 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = 𝑢 → (𝐹‘𝑤) = (𝐹‘𝑢)) |
| 51 | 50 | rspceeqv 3645 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑢 ∈ (𝐴 ∖ {𝑦}) ∧ (𝐹‘𝑢) = (𝐹‘𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)) |
| 52 | 49, 51 | mpan2 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 ∈ (𝐴 ∖ {𝑦}) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)) |
| 53 | 48, 52 | sylbir 235 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)) |
| 54 | 53 | adantll 714 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 ≠ 𝑦) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)) |
| 55 | 47, 54 | pm2.61dane 3029 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤)) |
| 56 | | fvres 6925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) = (𝐹‘𝑤)) |
| 57 | 56 | eqeq2d 2748 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → (𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ 𝑧 = (𝐹‘𝑤))) |
| 58 | 57 | rexbiia 3092 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑤 ∈
(𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹‘𝑤)) |
| 59 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = (𝐹‘𝑢) → (𝑧 = (𝐹‘𝑤) ↔ (𝐹‘𝑢) = (𝐹‘𝑤))) |
| 60 | 59 | rexbidv 3179 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝐹‘𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = (𝐹‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))) |
| 61 | 58, 60 | bitrid 283 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝐹‘𝑢) → (∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤) ↔ ∃𝑤 ∈ (𝐴 ∖ {𝑦})(𝐹‘𝑢) = (𝐹‘𝑤))) |
| 62 | 55, 61 | syl5ibrcom 247 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑢 ∈ 𝐴) → (𝑧 = (𝐹‘𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))) |
| 63 | 62 | rexlimdva 3155 |
. . . . . . . . . . . . . 14
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))) |
| 64 | 63 | imp 406 |
. . . . . . . . . . . . 13
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ ∃𝑢 ∈ 𝐴 𝑧 = (𝐹‘𝑢)) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)) |
| 65 | 33, 64 | syldan 591 |
. . . . . . . . . . . 12
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) ∧ 𝑧 ∈ 𝐵) → ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)) |
| 66 | 65 | ralrimiva 3146 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤)) |
| 67 | | dffo3 7122 |
. . . . . . . . . . 11
⊢ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto→𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ (𝐴 ∖ {𝑦})𝑧 = ((𝐹 ↾ (𝐴 ∖ {𝑦}))‘𝑤))) |
| 68 | 30, 66, 67 | sylanbrc 583 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto→𝐵) |
| 69 | | fodomfi 9350 |
. . . . . . . . . 10
⊢ (((𝐴 ∖ {𝑦}) ∈ Fin ∧ (𝐹 ↾ (𝐴 ∖ {𝑦})):(𝐴 ∖ {𝑦})–onto→𝐵) → 𝐵 ≼ (𝐴 ∖ {𝑦})) |
| 70 | 27, 68, 69 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))) → 𝐵 ≼ (𝐴 ∖ {𝑦})) |
| 71 | 70 | anassrs 467 |
. . . . . . . 8
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐵 ≼ (𝐴 ∖ {𝑦})) |
| 72 | 71 | expr 456 |
. . . . . . 7
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥 ≠ 𝑦 → 𝐵 ≼ (𝐴 ∖ {𝑦}))) |
| 73 | 72 | necon1bd 2958 |
. . . . . 6
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (¬ 𝐵 ≼ (𝐴 ∖ {𝑦}) → 𝑥 = 𝑦)) |
| 74 | 23, 73 | mpd 15 |
. . . . 5
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦) |
| 75 | 74 | ex 412 |
. . . 4
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
| 76 | 75 | ralrimivva 3202 |
. . 3
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
| 77 | | dff13 7275 |
. . 3
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
| 78 | 3, 76, 77 | sylanbrc 583 |
. 2
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1→𝐵) |
| 79 | | df-f1o 6568 |
. 2
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) |
| 80 | 78, 1, 79 | sylanbrc 583 |
1
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐵) |