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Theorem ordtbas2 23199
Description: Lemma for ordtbas 23200. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypotheses
Ref Expression
ordtval.1 𝑋 = dom 𝑅
ordtval.2 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
ordtval.3 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
ordtval.4 𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
Assertion
Ref Expression
ordtbas2 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
Distinct variable groups:   𝑎,𝑏,𝐴   𝑥,𝑎,𝑦,𝑅,𝑏   𝑋,𝑎,𝑏,𝑥,𝑦   𝐵,𝑎,𝑏
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ordtbas2
Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssun1 4178 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
2 ssun2 4179 . . . . . . 7 (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))
3 ordtval.1 . . . . . . . . . 10 𝑋 = dom 𝑅
4 ordtval.2 . . . . . . . . . 10 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
5 ordtval.3 . . . . . . . . . 10 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
63, 4, 5ordtuni 23198 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7923 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7eqeltrid 2845 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2842 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 7784 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 234 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 5323 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 587 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 ssexg 5323 . . . . . 6 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
151, 13, 14sylancr 587 . . . . 5 (𝑅 ∈ TosetRel → 𝐴 ∈ V)
16 ssun2 4179 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
17 ssexg 5323 . . . . . 6 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1816, 13, 17sylancr 587 . . . . 5 (𝑅 ∈ TosetRel → 𝐵 ∈ V)
19 elfiun 9470 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (fi‘(𝐴𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛))))
2015, 18, 19syl2anc 584 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛))))
213, 4ordtbaslem 23196 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
2221, 1eqsstrdi 4028 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ (𝐴𝐵))
23 ssun1 4178 . . . . . . 7 (𝐴𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶)
2422, 23sstrdi 3996 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ ((𝐴𝐵) ∪ 𝐶))
2524sseld 3982 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴𝐵) ∪ 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 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
2927, 28ordtbaslem 23196 . . . . . . . . . 10 (𝑅 ∈ TosetRel → (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
3026, 29syl 17 . . . . . . . . 9 (𝑅 ∈ TosetRel → (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
31 tsrps 18632 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
323psrn 18620 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
3331, 32syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅)
34 vex 3484 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
35 vex 3484 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
3634, 35brcnv 5893 . . . . . . . . . . . . . . . . 17 (𝑦𝑅𝑥𝑥𝑅𝑦)
3736bicomi 224 . . . . . . . . . . . . . . . 16 (𝑥𝑅𝑦𝑦𝑅𝑥)
3837notbii 320 . . . . . . . . . . . . . . 15 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥)
3938a1i 11 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
4033, 39rabeqbidv 3455 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
4133, 40mpteq12dv 5233 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4241rneqd 5949 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
435, 42eqtrid 2789 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4443fveq2d 6910 . . . . . . . . 9 (𝑅 ∈ TosetRel → (fi‘𝐵) = (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})))
4530, 44, 433eqtr4d 2787 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐵) = 𝐵)
4645, 16eqsstrdi 4028 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ (𝐴𝐵))
4746, 23sstrdi 3996 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶))
4847sseld 3982 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
49 ssun2 4179 . . . . . . . 8 𝐶 ⊆ ((𝐴𝐵) ∪ 𝐶)
5021, 4eqtrdi 2793 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐴) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
5150eleq2d 2827 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})))
52 breq2 5147 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑦𝑅𝑥𝑦𝑅𝑎))
5352notbid 318 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
5453rabbidv 3444 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5554cbvmptv 5255 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5655elrnmpt 5969 . . . . . . . . . . . . . . 15 (𝑚 ∈ V → (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
5756elv 3485 . . . . . . . . . . . . . 14 (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5851, 57bitrdi 287 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
5945, 5eqtrdi 2793 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐵) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
6059eleq2d 2827 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))
61 breq1 5146 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (𝑥𝑅𝑦𝑏𝑅𝑦))
6261notbid 318 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦))
6362rabbidv 3444 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6463cbvmptv 5255 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6564elrnmpt 5969 . . . . . . . . . . . . . . 15 (𝑛 ∈ V → (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
6665elv 3485 . . . . . . . . . . . . . 14 (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6760, 66bitrdi 287 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
6858, 67anbi12d 632 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})))
69 reeanv 3229 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
70 ineq12 4215 . . . . . . . . . . . . . . . 16 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
71 inrab 4316 . . . . . . . . . . . . . . . 16 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}
7270, 71eqtrdi 2793 . . . . . . . . . . . . . . 15 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7372reximi 3084 . . . . . . . . . . . . . 14 (∃𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7473reximi 3084 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7569, 74sylbir 235 . . . . . . . . . . . 12 ((∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7668, 75biimtrdi 253 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
7776imp 406 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
78 vex 3484 . . . . . . . . . . . 12 𝑚 ∈ V
7978inex1 5317 . . . . . . . . . . 11 (𝑚𝑛) ∈ V
80 eqid 2737 . . . . . . . . . . . 12 (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8180elrnmpog 7568 . . . . . . . . . . 11 ((𝑚𝑛) ∈ V → ((𝑚𝑛) ∈ ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
8279, 81ax-mp 5 . . . . . . . . . 10 ((𝑚𝑛) ∈ ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8377, 82sylibr 234 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
84 ordtval.4 . . . . . . . . 9 𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8583, 84eleqtrrdi 2852 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ 𝐶)
8649, 85sselid 3981 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶))
87 eleq1 2829 . . . . . . 7 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶)))
8886, 87syl5ibrcom 247 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
8988rexlimdvva 3213 . . . . 5 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9025, 48, 893jaod 1431 . . . 4 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9120, 90sylbid 240 . . 3 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9291ssrdv 3989 . 2 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ((𝐴𝐵) ∪ 𝐶))
93 ssfii 9459 . . . 4 ((𝐴𝐵) ∈ V → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9413, 93syl 17 . . 3 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9594adantr 480 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
96 simprl 771 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
97 eqidd 2738 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
9854rspceeqv 3645 . . . . . . . . . . . . . 14 ((𝑎𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
9996, 97, 98syl2anc 584 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1008adantr 480 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑋 ∈ V)
101 rabexg 5337 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
102 eqid 2737 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
103102elrnmpt 5969 . . . . . . . . . . . . . 14 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
104100, 101, 1033syl 18 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
10599, 104mpbird 257 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
106105, 4eleqtrrdi 2852 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴)
1071, 106sselid 3981 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴𝐵))
10895, 107sseldd 3984 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)))
109 simprr 773 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
110 eqidd 2738 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
11163rspceeqv 3645 . . . . . . . . . . . . . 14 ((𝑏𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
112109, 110, 111syl2anc 584 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
113 rabexg 5337 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V)
114 eqid 2737 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
115114elrnmpt 5969 . . . . . . . . . . . . . 14 ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V → ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
116100, 113, 1153syl 18 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
117112, 116mpbird 257 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
118117, 5eleqtrrdi 2852 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵)
11916, 118sselid 3981 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴𝐵))
12095, 119sseldd 3984 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵)))
121 fiin 9462 . . . . . . . . 9 (({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)) ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵))) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
122108, 120, 121syl2anc 584 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
12371, 122eqeltrrid 2846 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
124123ralrimivva 3202 . . . . . 6 (𝑅 ∈ TosetRel → ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
12580fmpo 8093 . . . . . 6 (∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)) ↔ (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
126124, 125sylib 218 . . . . 5 (𝑅 ∈ TosetRel → (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
127126frnd 6744 . . . 4 (𝑅 ∈ TosetRel → ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴𝐵)))
12884, 127eqsstrid 4022 . . 3 (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴𝐵)))
12994, 128unssd 4192 . 2 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴𝐵)))
13092, 129eqssd 4001 1 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3o 1086   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cun 3949  cin 3950  wss 3951  {csn 4626   cuni 4907   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  cmpo 7433  ficfi 9450  PosetRelcps 18609   TosetRel ctsr 18610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-ps 18611  df-tsr 18612
This theorem is referenced by:  ordtbas  23200  leordtval  23221
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