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Theorem ordtbas2 22702
Description: Lemma for ordtbas 22703. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypotheses
Ref Expression
ordtval.1 𝑋 = dom 𝑅
ordtval.2 𝐴 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
ordtval.3 𝐡 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
ordtval.4 𝐢 = ran (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
Assertion
Ref Expression
ordtbas2 (𝑅 ∈ TosetRel β†’ (fiβ€˜(𝐴 βˆͺ 𝐡)) = ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢))
Distinct variable groups:   π‘Ž,𝑏,𝐴   π‘₯,π‘Ž,𝑦,𝑅,𝑏   𝑋,π‘Ž,𝑏,π‘₯,𝑦   𝐡,π‘Ž,𝑏
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦)   𝐢(π‘₯,𝑦,π‘Ž,𝑏)

Proof of Theorem ordtbas2
Dummy variables π‘š 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssun1 4172 . . . . . 6 𝐴 βŠ† (𝐴 βˆͺ 𝐡)
2 ssun2 4173 . . . . . . 7 (𝐴 βˆͺ 𝐡) βŠ† ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡))
3 ordtval.1 . . . . . . . . . 10 𝑋 = dom 𝑅
4 ordtval.2 . . . . . . . . . 10 𝐴 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
5 ordtval.3 . . . . . . . . . 10 𝐡 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
63, 4, 5ordtuni 22701 . . . . . . . . 9 (𝑅 ∈ TosetRel β†’ 𝑋 = βˆͺ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)))
7 dmexg 7896 . . . . . . . . . 10 (𝑅 ∈ TosetRel β†’ dom 𝑅 ∈ V)
83, 7eqeltrid 2837 . . . . . . . . 9 (𝑅 ∈ TosetRel β†’ 𝑋 ∈ V)
96, 8eqeltrrd 2834 . . . . . . . 8 (𝑅 ∈ TosetRel β†’ βˆͺ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) ∈ V)
10 uniexb 7753 . . . . . . . 8 (({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) ∈ V ↔ βˆͺ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) ∈ V)
119, 10sylibr 233 . . . . . . 7 (𝑅 ∈ TosetRel β†’ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) ∈ V)
12 ssexg 5323 . . . . . . 7 (((𝐴 βˆͺ 𝐡) βŠ† ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) ∧ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) ∈ V) β†’ (𝐴 βˆͺ 𝐡) ∈ V)
132, 11, 12sylancr 587 . . . . . 6 (𝑅 ∈ TosetRel β†’ (𝐴 βˆͺ 𝐡) ∈ V)
14 ssexg 5323 . . . . . 6 ((𝐴 βŠ† (𝐴 βˆͺ 𝐡) ∧ (𝐴 βˆͺ 𝐡) ∈ V) β†’ 𝐴 ∈ V)
151, 13, 14sylancr 587 . . . . 5 (𝑅 ∈ TosetRel β†’ 𝐴 ∈ V)
16 ssun2 4173 . . . . . 6 𝐡 βŠ† (𝐴 βˆͺ 𝐡)
17 ssexg 5323 . . . . . 6 ((𝐡 βŠ† (𝐴 βˆͺ 𝐡) ∧ (𝐴 βˆͺ 𝐡) ∈ V) β†’ 𝐡 ∈ V)
1816, 13, 17sylancr 587 . . . . 5 (𝑅 ∈ TosetRel β†’ 𝐡 ∈ V)
19 elfiun 9427 . . . . 5 ((𝐴 ∈ V ∧ 𝐡 ∈ V) β†’ (𝑧 ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)) ↔ (𝑧 ∈ (fiβ€˜π΄) ∨ 𝑧 ∈ (fiβ€˜π΅) ∨ βˆƒπ‘š ∈ (fiβ€˜π΄)βˆƒπ‘› ∈ (fiβ€˜π΅)𝑧 = (π‘š ∩ 𝑛))))
2015, 18, 19syl2anc 584 . . . 4 (𝑅 ∈ TosetRel β†’ (𝑧 ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)) ↔ (𝑧 ∈ (fiβ€˜π΄) ∨ 𝑧 ∈ (fiβ€˜π΅) ∨ βˆƒπ‘š ∈ (fiβ€˜π΄)βˆƒπ‘› ∈ (fiβ€˜π΅)𝑧 = (π‘š ∩ 𝑛))))
213, 4ordtbaslem 22699 . . . . . . . 8 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΄) = 𝐴)
2221, 1eqsstrdi 4036 . . . . . . 7 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΄) βŠ† (𝐴 βˆͺ 𝐡))
23 ssun1 4172 . . . . . . 7 (𝐴 βˆͺ 𝐡) βŠ† ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)
2422, 23sstrdi 3994 . . . . . 6 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΄) βŠ† ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢))
2524sseld 3981 . . . . 5 (𝑅 ∈ TosetRel β†’ (𝑧 ∈ (fiβ€˜π΄) β†’ 𝑧 ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)))
26 cnvtsr 18543 . . . . . . . . . 10 (𝑅 ∈ TosetRel β†’ ◑𝑅 ∈ TosetRel )
27 df-rn 5687 . . . . . . . . . . 11 ran 𝑅 = dom ◑𝑅
28 eqid 2732 . . . . . . . . . . 11 ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) = ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯})
2927, 28ordtbaslem 22699 . . . . . . . . . 10 (◑𝑅 ∈ TosetRel β†’ (fiβ€˜ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯})) = ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}))
3026, 29syl 17 . . . . . . . . 9 (𝑅 ∈ TosetRel β†’ (fiβ€˜ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯})) = ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}))
31 tsrps 18542 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel β†’ 𝑅 ∈ PosetRel)
323psrn 18530 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel β†’ 𝑋 = ran 𝑅)
3331, 32syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel β†’ 𝑋 = ran 𝑅)
34 vex 3478 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
35 vex 3478 . . . . . . . . . . . . . . . . . 18 π‘₯ ∈ V
3634, 35brcnv 5882 . . . . . . . . . . . . . . . . 17 (𝑦◑𝑅π‘₯ ↔ π‘₯𝑅𝑦)
3736bicomi 223 . . . . . . . . . . . . . . . 16 (π‘₯𝑅𝑦 ↔ 𝑦◑𝑅π‘₯)
3837notbii 319 . . . . . . . . . . . . . . 15 (Β¬ π‘₯𝑅𝑦 ↔ Β¬ 𝑦◑𝑅π‘₯)
3938a1i 11 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel β†’ (Β¬ π‘₯𝑅𝑦 ↔ Β¬ 𝑦◑𝑅π‘₯))
4033, 39rabeqbidv 3449 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯})
4133, 40mpteq12dv 5239 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) = (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}))
4241rneqd 5937 . . . . . . . . . . 11 (𝑅 ∈ TosetRel β†’ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) = ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}))
435, 42eqtrid 2784 . . . . . . . . . 10 (𝑅 ∈ TosetRel β†’ 𝐡 = ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}))
4443fveq2d 6895 . . . . . . . . 9 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΅) = (fiβ€˜ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯})))
4530, 44, 433eqtr4d 2782 . . . . . . . 8 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΅) = 𝐡)
4645, 16eqsstrdi 4036 . . . . . . 7 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΅) βŠ† (𝐴 βˆͺ 𝐡))
4746, 23sstrdi 3994 . . . . . 6 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΅) βŠ† ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢))
4847sseld 3981 . . . . 5 (𝑅 ∈ TosetRel β†’ (𝑧 ∈ (fiβ€˜π΅) β†’ 𝑧 ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)))
49 ssun2 4173 . . . . . . . 8 𝐢 βŠ† ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)
5021, 4eqtrdi 2788 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΄) = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
5150eleq2d 2819 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel β†’ (π‘š ∈ (fiβ€˜π΄) ↔ π‘š ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))
52 breq2 5152 . . . . . . . . . . . . . . . . . . 19 (π‘₯ = π‘Ž β†’ (𝑦𝑅π‘₯ ↔ π‘¦π‘…π‘Ž))
5352notbid 317 . . . . . . . . . . . . . . . . . 18 (π‘₯ = π‘Ž β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ π‘¦π‘…π‘Ž))
5453rabbidv 3440 . . . . . . . . . . . . . . . . 17 (π‘₯ = π‘Ž β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
5554cbvmptv 5261 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (π‘Ž ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
5655elrnmpt 5955 . . . . . . . . . . . . . . 15 (π‘š ∈ V β†’ (π‘š ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) ↔ βˆƒπ‘Ž ∈ 𝑋 π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž}))
5756elv 3480 . . . . . . . . . . . . . 14 (π‘š ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) ↔ βˆƒπ‘Ž ∈ 𝑋 π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
5851, 57bitrdi 286 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel β†’ (π‘š ∈ (fiβ€˜π΄) ↔ βˆƒπ‘Ž ∈ 𝑋 π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž}))
5945, 5eqtrdi 2788 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΅) = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}))
6059eleq2d 2819 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel β†’ (𝑛 ∈ (fiβ€˜π΅) ↔ 𝑛 ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})))
61 breq1 5151 . . . . . . . . . . . . . . . . . . 19 (π‘₯ = 𝑏 β†’ (π‘₯𝑅𝑦 ↔ 𝑏𝑅𝑦))
6261notbid 317 . . . . . . . . . . . . . . . . . 18 (π‘₯ = 𝑏 β†’ (Β¬ π‘₯𝑅𝑦 ↔ Β¬ 𝑏𝑅𝑦))
6362rabbidv 3440 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑏 β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦})
6463cbvmptv 5261 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦})
6564elrnmpt 5955 . . . . . . . . . . . . . . 15 (𝑛 ∈ V β†’ (𝑛 ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) ↔ βˆƒπ‘ ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}))
6665elv 3480 . . . . . . . . . . . . . 14 (𝑛 ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) ↔ βˆƒπ‘ ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦})
6760, 66bitrdi 286 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel β†’ (𝑛 ∈ (fiβ€˜π΅) ↔ βˆƒπ‘ ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}))
6858, 67anbi12d 631 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel β†’ ((π‘š ∈ (fiβ€˜π΄) ∧ 𝑛 ∈ (fiβ€˜π΅)) ↔ (βˆƒπ‘Ž ∈ 𝑋 π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ βˆƒπ‘ ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦})))
69 reeanv 3226 . . . . . . . . . . . . 13 (βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) ↔ (βˆƒπ‘Ž ∈ 𝑋 π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ βˆƒπ‘ ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}))
70 ineq12 4207 . . . . . . . . . . . . . . . 16 ((π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) β†’ (π‘š ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}))
71 inrab 4306 . . . . . . . . . . . . . . . 16 ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}
7270, 71eqtrdi 2788 . . . . . . . . . . . . . . 15 ((π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) β†’ (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
7372reximi 3084 . . . . . . . . . . . . . 14 (βˆƒπ‘ ∈ 𝑋 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) β†’ βˆƒπ‘ ∈ 𝑋 (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
7473reximi 3084 . . . . . . . . . . . . 13 (βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) β†’ βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
7569, 74sylbir 234 . . . . . . . . . . . 12 ((βˆƒπ‘Ž ∈ 𝑋 π‘š = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∧ βˆƒπ‘ ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) β†’ βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
7668, 75syl6bi 252 . . . . . . . . . . 11 (𝑅 ∈ TosetRel β†’ ((π‘š ∈ (fiβ€˜π΄) ∧ 𝑛 ∈ (fiβ€˜π΅)) β†’ βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}))
7776imp 407 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (π‘š ∈ (fiβ€˜π΄) ∧ 𝑛 ∈ (fiβ€˜π΅))) β†’ βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
78 vex 3478 . . . . . . . . . . . 12 π‘š ∈ V
7978inex1 5317 . . . . . . . . . . 11 (π‘š ∩ 𝑛) ∈ V
80 eqid 2732 . . . . . . . . . . . 12 (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}) = (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
8180elrnmpog 7546 . . . . . . . . . . 11 ((π‘š ∩ 𝑛) ∈ V β†’ ((π‘š ∩ 𝑛) ∈ ran (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}) ↔ βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}))
8279, 81ax-mp 5 . . . . . . . . . 10 ((π‘š ∩ 𝑛) ∈ ran (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}) ↔ βˆƒπ‘Ž ∈ 𝑋 βˆƒπ‘ ∈ 𝑋 (π‘š ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
8377, 82sylibr 233 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (π‘š ∈ (fiβ€˜π΄) ∧ 𝑛 ∈ (fiβ€˜π΅))) β†’ (π‘š ∩ 𝑛) ∈ ran (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}))
84 ordtval.4 . . . . . . . . 9 𝐢 = ran (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)})
8583, 84eleqtrrdi 2844 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (π‘š ∈ (fiβ€˜π΄) ∧ 𝑛 ∈ (fiβ€˜π΅))) β†’ (π‘š ∩ 𝑛) ∈ 𝐢)
8649, 85sselid 3980 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (π‘š ∈ (fiβ€˜π΄) ∧ 𝑛 ∈ (fiβ€˜π΅))) β†’ (π‘š ∩ 𝑛) ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢))
87 eleq1 2821 . . . . . . 7 (𝑧 = (π‘š ∩ 𝑛) β†’ (𝑧 ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢) ↔ (π‘š ∩ 𝑛) ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)))
8886, 87syl5ibrcom 246 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (π‘š ∈ (fiβ€˜π΄) ∧ 𝑛 ∈ (fiβ€˜π΅))) β†’ (𝑧 = (π‘š ∩ 𝑛) β†’ 𝑧 ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)))
8988rexlimdvva 3211 . . . . 5 (𝑅 ∈ TosetRel β†’ (βˆƒπ‘š ∈ (fiβ€˜π΄)βˆƒπ‘› ∈ (fiβ€˜π΅)𝑧 = (π‘š ∩ 𝑛) β†’ 𝑧 ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)))
9025, 48, 893jaod 1428 . . . 4 (𝑅 ∈ TosetRel β†’ ((𝑧 ∈ (fiβ€˜π΄) ∨ 𝑧 ∈ (fiβ€˜π΅) ∨ βˆƒπ‘š ∈ (fiβ€˜π΄)βˆƒπ‘› ∈ (fiβ€˜π΅)𝑧 = (π‘š ∩ 𝑛)) β†’ 𝑧 ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)))
9120, 90sylbid 239 . . 3 (𝑅 ∈ TosetRel β†’ (𝑧 ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)) β†’ 𝑧 ∈ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢)))
9291ssrdv 3988 . 2 (𝑅 ∈ TosetRel β†’ (fiβ€˜(𝐴 βˆͺ 𝐡)) βŠ† ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢))
93 ssfii 9416 . . . 4 ((𝐴 βˆͺ 𝐡) ∈ V β†’ (𝐴 βˆͺ 𝐡) βŠ† (fiβ€˜(𝐴 βˆͺ 𝐡)))
9413, 93syl 17 . . 3 (𝑅 ∈ TosetRel β†’ (𝐴 βˆͺ 𝐡) βŠ† (fiβ€˜(𝐴 βˆͺ 𝐡)))
9594adantr 481 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (𝐴 βˆͺ 𝐡) βŠ† (fiβ€˜(𝐴 βˆͺ 𝐡)))
96 simprl 769 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ π‘Ž ∈ 𝑋)
97 eqidd 2733 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
9854rspceeqv 3633 . . . . . . . . . . . . . 14 ((π‘Ž ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž}) β†’ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
9996, 97, 98syl2anc 584 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
1008adantr 481 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ 𝑋 ∈ V)
101 rabexg 5331 . . . . . . . . . . . . . 14 (𝑋 ∈ V β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ V)
102 eqid 2732 . . . . . . . . . . . . . . 15 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
103102elrnmpt 5955 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ V β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) ↔ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
104100, 101, 1033syl 18 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) ↔ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
10599, 104mpbird 256 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
106105, 4eleqtrrdi 2844 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ 𝐴)
1071, 106sselid 3980 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ (𝐴 βˆͺ 𝐡))
10895, 107sseldd 3983 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)))
109 simprr 771 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ 𝑏 ∈ 𝑋)
110 eqidd 2733 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦})
11163rspceeqv 3633 . . . . . . . . . . . . . 14 ((𝑏 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) β†’ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
112109, 110, 111syl2anc 584 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
113 rabexg 5331 . . . . . . . . . . . . . 14 (𝑋 ∈ V β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ V)
114 eqid 2732 . . . . . . . . . . . . . . 15 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
115114elrnmpt 5955 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ V β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) ↔ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}))
116100, 113, 1153syl 18 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) ↔ βˆƒπ‘₯ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}))
117112, 116mpbird 256 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}))
118117, 5eleqtrrdi 2844 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ 𝐡)
11916, 118sselid 3980 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ (𝐴 βˆͺ 𝐡))
12095, 119sseldd 3983 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)))
121 fiin 9419 . . . . . . . . 9 (({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)) ∧ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦} ∈ (fiβ€˜(𝐴 βˆͺ 𝐡))) β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)))
122108, 120, 121syl2anc 584 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑏𝑅𝑦}) ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)))
12371, 122eqeltrrid 2838 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)} ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)))
124123ralrimivva 3200 . . . . . 6 (𝑅 ∈ TosetRel β†’ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)} ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)))
12580fmpo 8056 . . . . . 6 (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)} ∈ (fiβ€˜(𝐴 βˆͺ 𝐡)) ↔ (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}):(𝑋 Γ— 𝑋)⟢(fiβ€˜(𝐴 βˆͺ 𝐡)))
126124, 125sylib 217 . . . . 5 (𝑅 ∈ TosetRel β†’ (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}):(𝑋 Γ— 𝑋)⟢(fiβ€˜(𝐴 βˆͺ 𝐡)))
127126frnd 6725 . . . 4 (𝑅 ∈ TosetRel β†’ ran (π‘Ž ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑏𝑅𝑦)}) βŠ† (fiβ€˜(𝐴 βˆͺ 𝐡)))
12884, 127eqsstrid 4030 . . 3 (𝑅 ∈ TosetRel β†’ 𝐢 βŠ† (fiβ€˜(𝐴 βˆͺ 𝐡)))
12994, 128unssd 4186 . 2 (𝑅 ∈ TosetRel β†’ ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢) βŠ† (fiβ€˜(𝐴 βˆͺ 𝐡)))
13092, 129eqssd 3999 1 (𝑅 ∈ TosetRel β†’ (fiβ€˜(𝐴 βˆͺ 𝐡)) = ((𝐴 βˆͺ 𝐡) βˆͺ 𝐢))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  {csn 4628  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231   Γ— cxp 5674  β—‘ccnv 5675  dom cdm 5676  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543   ∈ cmpo 7413  ficfi 9407  PosetRelcps 18519   TosetRel ctsr 18520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-ps 18521  df-tsr 18522
This theorem is referenced by:  ordtbas  22703  leordtval  22724
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