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Theorem ordtbas2 23181
Description: Lemma for ordtbas 23182. (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 4171 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
2 ssun2 4172 . . . . . . 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 23180 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7904 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7eqeltrid 2830 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2827 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 7762 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 233 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 5319 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 585 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 ssexg 5319 . . . . . 6 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
151, 13, 14sylancr 585 . . . . 5 (𝑅 ∈ TosetRel → 𝐴 ∈ V)
16 ssun2 4172 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
17 ssexg 5319 . . . . . 6 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1816, 13, 17sylancr 585 . . . . 5 (𝑅 ∈ TosetRel → 𝐵 ∈ V)
19 elfiun 9464 . . . . 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 582 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛))))
213, 4ordtbaslem 23178 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
2221, 1eqsstrdi 4034 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ (𝐴𝐵))
23 ssun1 4171 . . . . . . 7 (𝐴𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶)
2422, 23sstrdi 3992 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ ((𝐴𝐵) ∪ 𝐶))
2524sseld 3978 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
26 cnvtsr 18606 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
27 df-rn 5684 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
28 eqid 2726 . . . . . . . . . . 11 ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
2927, 28ordtbaslem 23178 . . . . . . . . . 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 18605 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
323psrn 18593 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
3331, 32syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅)
34 vex 3467 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
35 vex 3467 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
3634, 35brcnv 5880 . . . . . . . . . . . . . . . . 17 (𝑦𝑅𝑥𝑥𝑅𝑦)
3736bicomi 223 . . . . . . . . . . . . . . . 16 (𝑥𝑅𝑦𝑦𝑅𝑥)
3837notbii 319 . . . . . . . . . . . . . . 15 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥)
3938a1i 11 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
4033, 39rabeqbidv 3438 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
4133, 40mpteq12dv 5235 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4241rneqd 5935 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
435, 42eqtrid 2778 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4443fveq2d 6895 . . . . . . . . 9 (𝑅 ∈ TosetRel → (fi‘𝐵) = (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})))
4530, 44, 433eqtr4d 2776 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐵) = 𝐵)
4645, 16eqsstrdi 4034 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ (𝐴𝐵))
4746, 23sstrdi 3992 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶))
4847sseld 3978 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
49 ssun2 4172 . . . . . . . 8 𝐶 ⊆ ((𝐴𝐵) ∪ 𝐶)
5021, 4eqtrdi 2782 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐴) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
5150eleq2d 2812 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})))
52 breq2 5148 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑦𝑅𝑥𝑦𝑅𝑎))
5352notbid 317 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
5453rabbidv 3428 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5554cbvmptv 5257 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5655elrnmpt 5953 . . . . . . . . . . . . . . 15 (𝑚 ∈ V → (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
5756elv 3469 . . . . . . . . . . . . . 14 (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5851, 57bitrdi 286 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
5945, 5eqtrdi 2782 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐵) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
6059eleq2d 2812 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))
61 breq1 5147 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (𝑥𝑅𝑦𝑏𝑅𝑦))
6261notbid 317 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦))
6362rabbidv 3428 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6463cbvmptv 5257 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6564elrnmpt 5953 . . . . . . . . . . . . . . 15 (𝑛 ∈ V → (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
6665elv 3469 . . . . . . . . . . . . . 14 (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6760, 66bitrdi 286 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
6858, 67anbi12d 630 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})))
69 reeanv 3217 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
70 ineq12 4206 . . . . . . . . . . . . . . . 16 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
71 inrab 4306 . . . . . . . . . . . . . . . 16 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}
7270, 71eqtrdi 2782 . . . . . . . . . . . . . . 15 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7372reximi 3074 . . . . . . . . . . . . . 14 (∃𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7473reximi 3074 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7569, 74sylbir 234 . . . . . . . . . . . 12 ((∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7668, 75biimtrdi 252 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
7776imp 405 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
78 vex 3467 . . . . . . . . . . . 12 𝑚 ∈ V
7978inex1 5313 . . . . . . . . . . 11 (𝑚𝑛) ∈ V
80 eqid 2726 . . . . . . . . . . . 12 (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8180elrnmpog 7551 . . . . . . . . . . 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 233 . . . . . . . . 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 2837 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ 𝐶)
8649, 85sselid 3977 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶))
87 eleq1 2814 . . . . . . 7 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶)))
8886, 87syl5ibrcom 246 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
8988rexlimdvva 3202 . . . . 5 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9025, 48, 893jaod 1426 . . . 4 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9120, 90sylbid 239 . . 3 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9291ssrdv 3985 . 2 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ((𝐴𝐵) ∪ 𝐶))
93 ssfii 9453 . . . 4 ((𝐴𝐵) ∈ V → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9413, 93syl 17 . . 3 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9594adantr 479 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
96 simprl 769 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
97 eqidd 2727 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
9854rspceeqv 3630 . . . . . . . . . . . . . 14 ((𝑎𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
9996, 97, 98syl2anc 582 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1008adantr 479 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑋 ∈ V)
101 rabexg 5329 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
102 eqid 2726 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
103102elrnmpt 5953 . . . . . . . . . . . . . 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 256 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
106105, 4eleqtrrdi 2837 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴)
1071, 106sselid 3977 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴𝐵))
10895, 107sseldd 3980 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)))
109 simprr 771 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
110 eqidd 2727 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
11163rspceeqv 3630 . . . . . . . . . . . . . 14 ((𝑏𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
112109, 110, 111syl2anc 582 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
113 rabexg 5329 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V)
114 eqid 2726 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
115114elrnmpt 5953 . . . . . . . . . . . . . 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 256 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
118117, 5eleqtrrdi 2837 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵)
11916, 118sselid 3977 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴𝐵))
12095, 119sseldd 3980 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵)))
121 fiin 9456 . . . . . . . . 9 (({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)) ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵))) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
122108, 120, 121syl2anc 582 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
12371, 122eqeltrrid 2831 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
124123ralrimivva 3191 . . . . . 6 (𝑅 ∈ TosetRel → ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
12580fmpo 8072 . . . . . 6 (∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)) ↔ (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
126124, 125sylib 217 . . . . 5 (𝑅 ∈ TosetRel → (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
127126frnd 6726 . . . 4 (𝑅 ∈ TosetRel → ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴𝐵)))
12884, 127eqsstrid 4028 . . 3 (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴𝐵)))
12994, 128unssd 4185 . 2 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴𝐵)))
13092, 129eqssd 3997 1 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3o 1083   = wceq 1534  wcel 2099  wral 3051  wrex 3060  {crab 3420  Vcvv 3463  cun 3945  cin 3946  wss 3947  {csn 4624   cuni 4906   class class class wbr 5144  cmpt 5227   × cxp 5671  ccnv 5672  dom cdm 5673  ran crn 5674  wf 6540  cfv 6544  cmpo 7416  ficfi 9444  PosetRelcps 18582   TosetRel ctsr 18583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-1o 8486  df-2o 8487  df-en 8965  df-fin 8968  df-fi 9445  df-ps 18584  df-tsr 18585
This theorem is referenced by:  ordtbas  23182  leordtval  23203
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