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Theorem ordtbas2 20905
Description: Lemma for ordtbas 20906. (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 3754 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
2 ssun2 3755 . . . . . . 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 20904 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7044 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7syl5eqel 2702 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2699 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 6921 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 224 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 4764 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 694 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 ssexg 4764 . . . . . 6 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
151, 13, 14sylancr 694 . . . . 5 (𝑅 ∈ TosetRel → 𝐴 ∈ V)
16 ssun2 3755 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
17 ssexg 4764 . . . . . 6 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1816, 13, 17sylancr 694 . . . . 5 (𝑅 ∈ TosetRel → 𝐵 ∈ V)
19 elfiun 8280 . . . . 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 692 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛))))
213, 4ordtbaslem 20902 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
2221, 1syl6eqss 3634 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ (𝐴𝐵))
23 ssun1 3754 . . . . . . 7 (𝐴𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶)
2422, 23syl6ss 3595 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ ((𝐴𝐵) ∪ 𝐶))
2524sseld 3582 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
26 cnvtsr 17143 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
27 df-rn 5085 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
28 eqid 2621 . . . . . . . . . . 11 ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
2927, 28ordtbaslem 20902 . . . . . . . . . 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 17142 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
323psrn 17130 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
3331, 32syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅)
34 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
35 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
3634, 35brcnv 5265 . . . . . . . . . . . . . . . . 17 (𝑦𝑅𝑥𝑥𝑅𝑦)
3736bicomi 214 . . . . . . . . . . . . . . . 16 (𝑥𝑅𝑦𝑦𝑅𝑥)
3837notbii 310 . . . . . . . . . . . . . . 15 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥)
3938a1i 11 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
4033, 39rabeqbidv 3181 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
4133, 40mpteq12dv 4693 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4241rneqd 5313 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
435, 42syl5eq 2667 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4443fveq2d 6152 . . . . . . . . 9 (𝑅 ∈ TosetRel → (fi‘𝐵) = (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})))
4530, 44, 433eqtr4d 2665 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐵) = 𝐵)
4645, 16syl6eqss 3634 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ (𝐴𝐵))
4746, 23syl6ss 3595 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶))
4847sseld 3582 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
49 ssun2 3755 . . . . . . . 8 𝐶 ⊆ ((𝐴𝐵) ∪ 𝐶)
5021, 4syl6eq 2671 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐴) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
5150eleq2d 2684 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})))
52 vex 3189 . . . . . . . . . . . . . . 15 𝑚 ∈ V
53 breq2 4617 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑦𝑅𝑥𝑦𝑅𝑎))
5453notbid 308 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
5554rabbidv 3177 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5655cbvmptv 4710 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5756elrnmpt 5332 . . . . . . . . . . . . . . 15 (𝑚 ∈ V → (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
5852, 57ax-mp 5 . . . . . . . . . . . . . 14 (𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5951, 58syl6bb 276 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
6045, 5syl6eq 2671 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐵) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
6160eleq2d 2684 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))
62 vex 3189 . . . . . . . . . . . . . . 15 𝑛 ∈ V
63 breq1 4616 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (𝑥𝑅𝑦𝑏𝑅𝑦))
6463notbid 308 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦))
6564rabbidv 3177 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6665cbvmptv 4710 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6766elrnmpt 5332 . . . . . . . . . . . . . . 15 (𝑛 ∈ V → (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
6862, 67ax-mp 5 . . . . . . . . . . . . . 14 (𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6961, 68syl6bb 276 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
7059, 69anbi12d 746 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})))
71 reeanv 3097 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
72 ineq12 3787 . . . . . . . . . . . . . . . 16 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
73 inrab 3875 . . . . . . . . . . . . . . . 16 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}
7472, 73syl6eq 2671 . . . . . . . . . . . . . . 15 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7574reximi 3005 . . . . . . . . . . . . . 14 (∃𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7675reximi 3005 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7771, 76sylbir 225 . . . . . . . . . . . 12 ((∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7870, 77syl6bi 243 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
7978imp 445 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8052inex1 4759 . . . . . . . . . . 11 (𝑚𝑛) ∈ V
81 eqid 2621 . . . . . . . . . . . 12 (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8281elrnmpt2g 6725 . . . . . . . . . . 11 ((𝑚𝑛) ∈ V → ((𝑚𝑛) ∈ ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
8380, 82ax-mp 5 . . . . . . . . . 10 ((𝑚𝑛) ∈ ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8479, 83sylibr 224 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
85 ordtval.4 . . . . . . . . 9 𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8684, 85syl6eleqr 2709 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ 𝐶)
8749, 86sseldi 3581 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶))
88 eleq1 2686 . . . . . . 7 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶)))
8987, 88syl5ibrcom 237 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9089rexlimdvva 3031 . . . . 5 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9125, 48, 903jaod 1389 . . . 4 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9220, 91sylbid 230 . . 3 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9392ssrdv 3589 . 2 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ((𝐴𝐵) ∪ 𝐶))
94 ssfii 8269 . . . 4 ((𝐴𝐵) ∈ V → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9513, 94syl 17 . . 3 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9695adantr 481 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
97 simprl 793 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
98 eqidd 2622 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
9955eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
10099rspcev 3295 . . . . . . . . . . . . . 14 ((𝑎𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
10197, 98, 100syl2anc 692 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1028adantr 481 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑋 ∈ V)
103 rabexg 4772 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
104 eqid 2621 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
105104elrnmpt 5332 . . . . . . . . . . . . . 14 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
106102, 103, 1053syl 18 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
107101, 106mpbird 247 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
108107, 4syl6eleqr 2709 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴)
1091, 108sseldi 3581 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴𝐵))
11096, 109sseldd 3584 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)))
111 simprr 795 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
112 eqidd 2622 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
11365eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑥 = 𝑏 → ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ↔ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
114113rspcev 3295 . . . . . . . . . . . . . 14 ((𝑏𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
115111, 112, 114syl2anc 692 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
116 rabexg 4772 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V)
117 eqid 2621 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
118117elrnmpt 5332 . . . . . . . . . . . . . 14 ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V → ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
119102, 116, 1183syl 18 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
120115, 119mpbird 247 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
121120, 5syl6eleqr 2709 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵)
12216, 121sseldi 3581 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴𝐵))
12396, 122sseldd 3584 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵)))
124 fiin 8272 . . . . . . . . 9 (({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)) ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵))) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
125110, 123, 124syl2anc 692 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
12673, 125syl5eqelr 2703 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
127126ralrimivva 2965 . . . . . 6 (𝑅 ∈ TosetRel → ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
12881fmpt2 7182 . . . . . 6 (∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)) ↔ (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
129127, 128sylib 208 . . . . 5 (𝑅 ∈ TosetRel → (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
130 frn 6010 . . . . 5 ((𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)) → ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴𝐵)))
131129, 130syl 17 . . . 4 (𝑅 ∈ TosetRel → ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴𝐵)))
13285, 131syl5eqss 3628 . . 3 (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴𝐵)))
13395, 132unssd 3767 . 2 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴𝐵)))
13493, 133eqssd 3600 1 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cun 3553  cin 3554  wss 3555  {csn 4148   cuni 4402   class class class wbr 4613  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  wf 5843  cfv 5847  cmpt2 6606  ficfi 8260  PosetRelcps 17119   TosetRel ctsr 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-ps 17121  df-tsr 17122
This theorem is referenced by:  ordtbas  20906  leordtval  20927
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