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Theorem ordtbas2 21806
 Description: Lemma for ordtbas 21807. (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 4099 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
2 ssun2 4100 . . . . . . 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 21805 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7597 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7eqeltrid 2894 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2891 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 7469 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 237 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 5192 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 590 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 ssexg 5192 . . . . . 6 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
151, 13, 14sylancr 590 . . . . 5 (𝑅 ∈ TosetRel → 𝐴 ∈ V)
16 ssun2 4100 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
17 ssexg 5192 . . . . . 6 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1816, 13, 17sylancr 590 . . . . 5 (𝑅 ∈ TosetRel → 𝐵 ∈ V)
19 elfiun 8881 . . . . 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 587 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛))))
213, 4ordtbaslem 21803 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
2221, 1eqsstrdi 3969 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ (𝐴𝐵))
23 ssun1 4099 . . . . . . 7 (𝐴𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶)
2422, 23sstrdi 3927 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ ((𝐴𝐵) ∪ 𝐶))
2524sseld 3914 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
26 cnvtsr 17827 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
27 df-rn 5531 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
28 eqid 2798 . . . . . . . . . . 11 ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
2927, 28ordtbaslem 21803 . . . . . . . . . 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 17826 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
323psrn 17814 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
3331, 32syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅)
34 vex 3444 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
35 vex 3444 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
3634, 35brcnv 5718 . . . . . . . . . . . . . . . . 17 (𝑦𝑅𝑥𝑥𝑅𝑦)
3736bicomi 227 . . . . . . . . . . . . . . . 16 (𝑥𝑅𝑦𝑦𝑅𝑥)
3837notbii 323 . . . . . . . . . . . . . . 15 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥)
3938a1i 11 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
4033, 39rabeqbidv 3433 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
4133, 40mpteq12dv 5116 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4241rneqd 5773 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
435, 42syl5eq 2845 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4443fveq2d 6650 . . . . . . . . 9 (𝑅 ∈ TosetRel → (fi‘𝐵) = (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})))
4530, 44, 433eqtr4d 2843 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐵) = 𝐵)
4645, 16eqsstrdi 3969 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ (𝐴𝐵))
4746, 23sstrdi 3927 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶))
4847sseld 3914 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
49 ssun2 4100 . . . . . . . 8 𝐶 ⊆ ((𝐴𝐵) ∪ 𝐶)
5021, 4eqtrdi 2849 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐴) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
5150eleq2d 2875 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})))
52 breq2 5035 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑦𝑅𝑥𝑦𝑅𝑎))
5352notbid 321 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
5453rabbidv 3427 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5554cbvmptv 5134 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5655elrnmpt 5793 . . . . . . . . . . . . . . 15 (𝑚 ∈ V → (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
5756elv 3446 . . . . . . . . . . . . . 14 (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5851, 57syl6bb 290 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
5945, 5eqtrdi 2849 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐵) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
6059eleq2d 2875 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))
61 breq1 5034 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (𝑥𝑅𝑦𝑏𝑅𝑦))
6261notbid 321 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦))
6362rabbidv 3427 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6463cbvmptv 5134 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6564elrnmpt 5793 . . . . . . . . . . . . . . 15 (𝑛 ∈ V → (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
6665elv 3446 . . . . . . . . . . . . . 14 (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6760, 66syl6bb 290 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
6858, 67anbi12d 633 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})))
69 reeanv 3320 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
70 ineq12 4134 . . . . . . . . . . . . . . . 16 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
71 inrab 4227 . . . . . . . . . . . . . . . 16 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}
7270, 71eqtrdi 2849 . . . . . . . . . . . . . . 15 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7372reximi 3206 . . . . . . . . . . . . . 14 (∃𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7473reximi 3206 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7569, 74sylbir 238 . . . . . . . . . . . 12 ((∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7668, 75syl6bi 256 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
7776imp 410 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
78 vex 3444 . . . . . . . . . . . 12 𝑚 ∈ V
7978inex1 5186 . . . . . . . . . . 11 (𝑚𝑛) ∈ V
80 eqid 2798 . . . . . . . . . . . 12 (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8180elrnmpog 7267 . . . . . . . . . . 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 237 . . . . . . . . 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 2901 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ 𝐶)
8649, 85sseldi 3913 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶))
87 eleq1 2877 . . . . . . 7 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶)))
8886, 87syl5ibrcom 250 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
8988rexlimdvva 3253 . . . . 5 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9025, 48, 893jaod 1425 . . . 4 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9120, 90sylbid 243 . . 3 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9291ssrdv 3921 . 2 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ((𝐴𝐵) ∪ 𝐶))
93 ssfii 8870 . . . 4 ((𝐴𝐵) ∈ V → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9413, 93syl 17 . . 3 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9594adantr 484 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
96 simprl 770 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
97 eqidd 2799 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
9854rspceeqv 3586 . . . . . . . . . . . . . 14 ((𝑎𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
9996, 97, 98syl2anc 587 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1008adantr 484 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑋 ∈ V)
101 rabexg 5199 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
102 eqid 2798 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
103102elrnmpt 5793 . . . . . . . . . . . . . 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 260 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
106105, 4eleqtrrdi 2901 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴)
1071, 106sseldi 3913 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴𝐵))
10895, 107sseldd 3916 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)))
109 simprr 772 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
110 eqidd 2799 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
11163rspceeqv 3586 . . . . . . . . . . . . . 14 ((𝑏𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
112109, 110, 111syl2anc 587 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
113 rabexg 5199 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V)
114 eqid 2798 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
115114elrnmpt 5793 . . . . . . . . . . . . . 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 260 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
118117, 5eleqtrrdi 2901 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵)
11916, 118sseldi 3913 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴𝐵))
12095, 119sseldd 3916 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵)))
121 fiin 8873 . . . . . . . . 9 (({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)) ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵))) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
122108, 120, 121syl2anc 587 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
12371, 122eqeltrrid 2895 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
124123ralrimivva 3156 . . . . . 6 (𝑅 ∈ TosetRel → ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
12580fmpo 7751 . . . . . 6 (∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)) ↔ (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
126124, 125sylib 221 . . . . 5 (𝑅 ∈ TosetRel → (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
127126frnd 6495 . . . 4 (𝑅 ∈ TosetRel → ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴𝐵)))
12884, 127eqsstrid 3963 . . 3 (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴𝐵)))
12994, 128unssd 4113 . 2 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴𝐵)))
13092, 129eqssd 3932 1 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  {csn 4525  ∪ cuni 4801   class class class wbr 5031   ↦ cmpt 5111   × cxp 5518  ◡ccnv 5519  dom cdm 5520  ran crn 5521  ⟶wf 6321  ‘cfv 6325   ∈ cmpo 7138  ficfi 8861  PosetRelcps 17803   TosetRel ctsr 17804 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-en 8496  df-fin 8499  df-fi 8862  df-ps 17805  df-tsr 17806 This theorem is referenced by:  ordtbas  21807  leordtval  21828
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