| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ssun1 4178 | . . . . . 6
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | 
| 2 |  | ssun2 4179 | . . . . . . 7
⊢ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) | 
| 3 |  | ordtval.1 | . . . . . . . . . 10
⊢ 𝑋 = dom 𝑅 | 
| 4 |  | ordtval.2 | . . . . . . . . . 10
⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) | 
| 5 |  | ordtval.3 | . . . . . . . . . 10
⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) | 
| 6 | 3, 4, 5 | ordtuni 23198 | . . . . . . . . 9
⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪
({𝑋} ∪ (𝐴 ∪ 𝐵))) | 
| 7 |  | dmexg 7923 | . . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → dom
𝑅 ∈
V) | 
| 8 | 3, 7 | eqeltrid 2845 | . . . . . . . . 9
⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V) | 
| 9 | 6, 8 | eqeltrrd 2842 | . . . . . . . 8
⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋}
∪ (𝐴 ∪ 𝐵)) ∈ V) | 
| 10 |  | uniexb 7784 | . . . . . . . 8
⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋}
∪ (𝐴 ∪ 𝐵)) ∈ V) | 
| 11 | 9, 10 | sylibr 234 | . . . . . . 7
⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) | 
| 12 |  | ssexg 5323 | . . . . . . 7
⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | 
| 13 | 2, 11, 12 | sylancr 587 | . . . . . 6
⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V) | 
| 14 |  | ssexg 5323 | . . . . . 6
⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) | 
| 15 | 1, 13, 14 | sylancr 587 | . . . . 5
⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V) | 
| 16 |  | ssun2 4179 | . . . . . 6
⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | 
| 17 |  | ssexg 5323 | . . . . . 6
⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) | 
| 18 | 16, 13, 17 | sylancr 587 | . . . . 5
⊢ (𝑅 ∈ TosetRel → 𝐵 ∈ V) | 
| 19 |  | elfiun 9470 | . . . . 5
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)))) | 
| 20 | 15, 18, 19 | syl2anc 584 | . . . 4
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)))) | 
| 21 | 3, 4 | ordtbaslem 23196 | . . . . . . . 8
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) = 𝐴) | 
| 22 | 21, 1 | eqsstrdi 4028 | . . . . . . 7
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) ⊆ (𝐴 ∪ 𝐵)) | 
| 23 |  | ssun1 4178 | . . . . . . 7
⊢ (𝐴 ∪ 𝐵) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶) | 
| 24 | 22, 23 | sstrdi 3996 | . . . . . 6
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) ⊆
((𝐴 ∪ 𝐵) ∪ 𝐶)) | 
| 25 | 24 | sseld 3982 | . . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) | 
| 26 |  | cnvtsr 18633 | . . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) | 
| 27 |  | df-rn 5696 | . . . . . . . . . . 11
⊢ ran 𝑅 = dom ◡𝑅 | 
| 28 |  | eqid 2737 | . . . . . . . . . . 11
⊢ ran
(𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) | 
| 29 | 27, 28 | ordtbaslem 23196 | . . . . . . . . . 10
⊢ (◡𝑅 ∈ TosetRel → (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) | 
| 30 | 26, 29 | syl 17 | . . . . . . . . 9
⊢ (𝑅 ∈ TosetRel →
(fi‘ran (𝑥 ∈ ran
𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) | 
| 31 |  | tsrps 18632 | . . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈
PosetRel) | 
| 32 | 3 | psrn 18620 | . . . . . . . . . . . . . 14
⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) | 
| 33 | 31, 32 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅) | 
| 34 |  | vex 3484 | . . . . . . . . . . . . . . . . . 18
⊢ 𝑦 ∈ V | 
| 35 |  | vex 3484 | . . . . . . . . . . . . . . . . . 18
⊢ 𝑥 ∈ V | 
| 36 | 34, 35 | brcnv 5893 | . . . . . . . . . . . . . . . . 17
⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) | 
| 37 | 36 | bicomi 224 | . . . . . . . . . . . . . . . 16
⊢ (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥) | 
| 38 | 37 | notbii 320 | . . . . . . . . . . . . . . 15
⊢ (¬
𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥) | 
| 39 | 38 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (¬
𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥)) | 
| 40 | 33, 39 | rabeqbidv 3455 | . . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) | 
| 41 | 33, 40 | mpteq12dv 5233 | . . . . . . . . . . . 12
⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) | 
| 42 | 41 | rneqd 5949 | . . . . . . . . . . 11
⊢ (𝑅 ∈ TosetRel → ran
(𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) | 
| 43 | 5, 42 | eqtrid 2789 | . . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) | 
| 44 | 43 | fveq2d 6910 | . . . . . . . . 9
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) =
(fi‘ran (𝑥 ∈ ran
𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))) | 
| 45 | 30, 44, 43 | 3eqtr4d 2787 | . . . . . . . 8
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) = 𝐵) | 
| 46 | 45, 16 | eqsstrdi 4028 | . . . . . . 7
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) ⊆ (𝐴 ∪ 𝐵)) | 
| 47 | 46, 23 | sstrdi 3996 | . . . . . 6
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) ⊆
((𝐴 ∪ 𝐵) ∪ 𝐶)) | 
| 48 | 47 | sseld 3982 | . . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) | 
| 49 |  | ssun2 4179 | . . . . . . . 8
⊢ 𝐶 ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶) | 
| 50 | 21, 4 | eqtrdi 2793 | . . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) | 
| 51 | 50 | eleq2d 2827 | . . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) | 
| 52 |  | breq2 5147 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎)) | 
| 53 | 52 | notbid 318 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎)) | 
| 54 | 53 | rabbidv 3444 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) | 
| 55 | 54 | cbvmptv 5255 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) | 
| 56 | 55 | elrnmpt 5969 | . . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ V → (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})) | 
| 57 | 56 | elv 3485 | . . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) | 
| 58 | 51, 57 | bitrdi 287 | . . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})) | 
| 59 | 45, 5 | eqtrdi 2793 | . . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) | 
| 60 | 59 | eleq2d 2827 | . . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) | 
| 61 |  | breq1 5146 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑏 → (𝑥𝑅𝑦 ↔ 𝑏𝑅𝑦)) | 
| 62 | 61 | notbid 318 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦)) | 
| 63 | 62 | rabbidv 3444 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) | 
| 64 | 63 | cbvmptv 5255 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) | 
| 65 | 64 | elrnmpt 5969 | . . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) | 
| 66 | 65 | elv 3485 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) | 
| 67 | 60, 66 | bitrdi 287 | . . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) | 
| 68 | 58, 67 | anbi12d 632 | . . . . . . . . . . . 12
⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))) | 
| 69 |  | reeanv 3229 | . . . . . . . . . . . . 13
⊢
(∃𝑎 ∈
𝑋 ∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) | 
| 70 |  | ineq12 4215 | . . . . . . . . . . . . . . . 16
⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) | 
| 71 |  | inrab 4316 | . . . . . . . . . . . . . . . 16
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} | 
| 72 | 70, 71 | eqtrdi 2793 | . . . . . . . . . . . . . . 15
⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 73 | 72 | reximi 3084 | . . . . . . . . . . . . . 14
⊢
(∃𝑏 ∈
𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 74 | 73 | reximi 3084 | . . . . . . . . . . . . 13
⊢
(∃𝑎 ∈
𝑋 ∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 75 | 69, 74 | sylbir 235 | . . . . . . . . . . . 12
⊢
((∃𝑎 ∈
𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 76 | 68, 75 | biimtrdi 253 | . . . . . . . . . . 11
⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) | 
| 77 | 76 | imp 406 | . . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 78 |  | vex 3484 | . . . . . . . . . . . 12
⊢ 𝑚 ∈ V | 
| 79 | 78 | inex1 5317 | . . . . . . . . . . 11
⊢ (𝑚 ∩ 𝑛) ∈ V | 
| 80 |  | eqid 2737 | . . . . . . . . . . . 12
⊢ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 81 | 80 | elrnmpog 7568 | . . . . . . . . . . 11
⊢ ((𝑚 ∩ 𝑛) ∈ V → ((𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) | 
| 82 | 79, 81 | ax-mp 5 | . . . . . . . . . 10
⊢ ((𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 83 | 77, 82 | sylibr 234 | . . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) | 
| 84 |  | ordtval.4 | . . . . . . . . 9
⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) | 
| 85 | 83, 84 | eleqtrrdi 2852 | . . . . . . . 8
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ 𝐶) | 
| 86 | 49, 85 | sselid 3981 | . . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)) | 
| 87 |  | eleq1 2829 | . . . . . . 7
⊢ (𝑧 = (𝑚 ∩ 𝑛) → (𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶) ↔ (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) | 
| 88 | 86, 87 | syl5ibrcom 247 | . . . . . 6
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) | 
| 89 | 88 | rexlimdvva 3213 | . . . . 5
⊢ (𝑅 ∈ TosetRel →
(∃𝑚 ∈
(fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) | 
| 90 | 25, 48, 89 | 3jaod 1431 | . . . 4
⊢ (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) | 
| 91 | 20, 90 | sylbid 240 | . . 3
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) | 
| 92 | 91 | ssrdv 3989 | . 2
⊢ (𝑅 ∈ TosetRel →
(fi‘(𝐴 ∪ 𝐵)) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶)) | 
| 93 |  | ssfii 9459 | . . . 4
⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) | 
| 94 | 13, 93 | syl 17 | . . 3
⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) | 
| 95 | 94 | adantr 480 | . . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) | 
| 96 |  | simprl 771 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋) | 
| 97 |  | eqidd 2738 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) | 
| 98 | 54 | rspceeqv 3645 | . . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) | 
| 99 | 96, 97, 98 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) | 
| 100 | 8 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V) | 
| 101 |  | rabexg 5337 | . . . . . . . . . . . . . 14
⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V) | 
| 102 |  | eqid 2737 | . . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) | 
| 103 | 102 | elrnmpt 5969 | . . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) | 
| 104 | 100, 101,
103 | 3syl 18 | . . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) | 
| 105 | 99, 104 | mpbird 257 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) | 
| 106 | 105, 4 | eleqtrrdi 2852 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴) | 
| 107 | 1, 106 | sselid 3981 | . . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴 ∪ 𝐵)) | 
| 108 | 95, 107 | sseldd 3984 | . . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵))) | 
| 109 |  | simprr 773 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋) | 
| 110 |  | eqidd 2738 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) | 
| 111 | 63 | rspceeqv 3645 | . . . . . . . . . . . . . 14
⊢ ((𝑏 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) | 
| 112 | 109, 110,
111 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) | 
| 113 |  | rabexg 5337 | . . . . . . . . . . . . . 14
⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V) | 
| 114 |  | eqid 2737 | . . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) | 
| 115 | 114 | elrnmpt 5969 | . . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) | 
| 116 | 100, 113,
115 | 3syl 18 | . . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) | 
| 117 | 112, 116 | mpbird 257 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) | 
| 118 | 117, 5 | eleqtrrdi 2852 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵) | 
| 119 | 16, 118 | sselid 3981 | . . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴 ∪ 𝐵)) | 
| 120 | 95, 119 | sseldd 3984 | . . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵))) | 
| 121 |  | fiin 9462 | . . . . . . . . 9
⊢ (({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵)) ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵))) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵))) | 
| 122 | 108, 120,
121 | syl2anc 584 | . . . . . . . 8
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵))) | 
| 123 | 71, 122 | eqeltrrid 2846 | . . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵))) | 
| 124 | 123 | ralrimivva 3202 | . . . . . 6
⊢ (𝑅 ∈ TosetRel →
∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵))) | 
| 125 | 80 | fmpo 8093 | . . . . . 6
⊢
(∀𝑎 ∈
𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵))) | 
| 126 | 124, 125 | sylib 218 | . . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵))) | 
| 127 | 126 | frnd 6744 | . . . 4
⊢ (𝑅 ∈ TosetRel → ran
(𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴 ∪ 𝐵))) | 
| 128 | 84, 127 | eqsstrid 4022 | . . 3
⊢ (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴 ∪ 𝐵))) | 
| 129 | 94, 128 | unssd 4192 | . 2
⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴 ∪ 𝐵))) | 
| 130 | 92, 129 | eqssd 4001 | 1
⊢ (𝑅 ∈ TosetRel →
(fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)) |