Proof of Theorem mapsnd
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mapsnd.1 | . . . 4
⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| 2 |  | snex 5436 | . . . . 5
⊢ {𝐵} ∈ V | 
| 3 | 2 | a1i 11 | . . . 4
⊢ (𝜑 → {𝐵} ∈ V) | 
| 4 | 1, 3 | elmapd 8880 | . . 3
⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)) | 
| 5 |  | ffn 6736 | . . . . . . . . 9
⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵}) | 
| 6 |  | mapsnd.2 | . . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ 𝑊) | 
| 7 |  | snidg 4660 | . . . . . . . . . 10
⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵}) | 
| 8 | 6, 7 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ {𝐵}) | 
| 9 |  | fneu 6678 | . . . . . . . . 9
⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦) | 
| 10 | 5, 8, 9 | syl2anr 597 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦) | 
| 11 |  | euabsn 4726 | . . . . . . . . . 10
⊢
(∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦}) | 
| 12 |  | frel 6741 | . . . . . . . . . . . . . 14
⊢ (𝑓:{𝐵}⟶𝐴 → Rel 𝑓) | 
| 13 |  | relimasn 6103 | . . . . . . . . . . . . . 14
⊢ (Rel
𝑓 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦}) | 
| 14 | 12, 13 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦}) | 
| 15 |  | fdm 6745 | . . . . . . . . . . . . . . 15
⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵}) | 
| 16 | 15 | imaeq2d 6078 | . . . . . . . . . . . . . 14
⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵})) | 
| 17 |  | imadmrn 6088 | . . . . . . . . . . . . . 14
⊢ (𝑓 “ dom 𝑓) = ran 𝑓 | 
| 18 | 16, 17 | eqtr3di 2792 | . . . . . . . . . . . . 13
⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓) | 
| 19 | 14, 18 | eqtr3d 2779 | . . . . . . . . . . . 12
⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓) | 
| 20 | 19 | eqeq1d 2739 | . . . . . . . . . . 11
⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦})) | 
| 21 | 20 | exbidv 1921 | . . . . . . . . . 10
⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦})) | 
| 22 | 11, 21 | bitrid 283 | . . . . . . . . 9
⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦})) | 
| 23 | 22 | adantl 481 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦})) | 
| 24 | 10, 23 | mpbid 232 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦}) | 
| 25 |  | frn 6743 | . . . . . . . . . . . . 13
⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴) | 
| 26 | 25 | sseld 3982 | . . . . . . . . . . . 12
⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴)) | 
| 27 |  | vsnid 4663 | . . . . . . . . . . . . 13
⊢ 𝑦 ∈ {𝑦} | 
| 28 |  | eleq2 2830 | . . . . . . . . . . . . 13
⊢ (ran
𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦})) | 
| 29 | 27, 28 | mpbiri 258 | . . . . . . . . . . . 12
⊢ (ran
𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓) | 
| 30 | 26, 29 | impel 505 | . . . . . . . . . . 11
⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴) | 
| 31 | 30 | adantll 714 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴) | 
| 32 |  | ffrn 6749 | . . . . . . . . . . . . . 14
⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓) | 
| 33 |  | feq3 6718 | . . . . . . . . . . . . . 14
⊢ (ran
𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦})) | 
| 34 | 32, 33 | syl5ibcom 245 | . . . . . . . . . . . . 13
⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦})) | 
| 35 | 34 | imp 406 | . . . . . . . . . . . 12
⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦}) | 
| 36 | 35 | adantll 714 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦}) | 
| 37 | 6 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵 ∈ 𝑊) | 
| 38 |  | vex 3484 | . . . . . . . . . . . 12
⊢ 𝑦 ∈ V | 
| 39 |  | fsng 7157 | . . . . . . . . . . . 12
⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {〈𝐵, 𝑦〉})) | 
| 40 | 37, 38, 39 | sylancl 586 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {〈𝐵, 𝑦〉})) | 
| 41 | 36, 40 | mpbid 232 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {〈𝐵, 𝑦〉}) | 
| 42 | 31, 41 | jca 511 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉})) | 
| 43 | 42 | ex 412 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉}))) | 
| 44 | 43 | eximdv 1917 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉}))) | 
| 45 | 24, 44 | mpd 15 | . . . . . 6
⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉})) | 
| 46 |  | df-rex 3071 | . . . . . 6
⊢
(∃𝑦 ∈
𝐴 𝑓 = {〈𝐵, 𝑦〉} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉})) | 
| 47 | 45, 46 | sylibr 234 | . . . . 5
⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}) | 
| 48 | 47 | ex 412 | . . . 4
⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉})) | 
| 49 |  | f1osng 6889 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → {〈𝐵, 𝑦〉}:{𝐵}–1-1-onto→{𝑦}) | 
| 50 | 6, 38, 49 | sylancl 586 | . . . . . . . . . 10
⊢ (𝜑 → {〈𝐵, 𝑦〉}:{𝐵}–1-1-onto→{𝑦}) | 
| 51 | 50 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 = {〈𝐵, 𝑦〉}) → {〈𝐵, 𝑦〉}:{𝐵}–1-1-onto→{𝑦}) | 
| 52 |  | f1oeq1 6836 | . . . . . . . . . . 11
⊢ (𝑓 = {〈𝐵, 𝑦〉} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {〈𝐵, 𝑦〉}:{𝐵}–1-1-onto→{𝑦})) | 
| 53 | 52 | bicomd 223 | . . . . . . . . . 10
⊢ (𝑓 = {〈𝐵, 𝑦〉} → ({〈𝐵, 𝑦〉}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦})) | 
| 54 | 53 | adantl 481 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 = {〈𝐵, 𝑦〉}) → ({〈𝐵, 𝑦〉}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦})) | 
| 55 | 51, 54 | mpbid 232 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 = {〈𝐵, 𝑦〉}) → 𝑓:{𝐵}–1-1-onto→{𝑦}) | 
| 56 |  | f1of 6848 | . . . . . . . 8
⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦}) | 
| 57 | 55, 56 | syl 17 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓 = {〈𝐵, 𝑦〉}) → 𝑓:{𝐵}⟶{𝑦}) | 
| 58 | 57 | 3adant2 1132 | . . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉}) → 𝑓:{𝐵}⟶{𝑦}) | 
| 59 |  | snssi 4808 | . . . . . . 7
⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴) | 
| 60 | 59 | 3ad2ant2 1135 | . . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉}) → {𝑦} ⊆ 𝐴) | 
| 61 | 58, 60 | fssd 6753 | . . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {〈𝐵, 𝑦〉}) → 𝑓:{𝐵}⟶𝐴) | 
| 62 | 61 | rexlimdv3a 3159 | . . . 4
⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉} → 𝑓:{𝐵}⟶𝐴)) | 
| 63 | 48, 62 | impbid 212 | . . 3
⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉})) | 
| 64 | 4, 63 | bitrd 279 | . 2
⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉})) | 
| 65 | 64 | eqabdv 2875 | 1
⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}}) |