MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtbas2 Structured version   Visualization version   GIF version

Theorem ordtbas2 20805
Description: Lemma for ordtbas 20806. (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 3738 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
2 ssun2 3739 . . . . . . 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 20804 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 6989 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7syl5eqel 2692 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2689 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 6866 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 223 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 4732 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 694 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 ssexg 4732 . . . . . 6 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
151, 13, 14sylancr 694 . . . . 5 (𝑅 ∈ TosetRel → 𝐴 ∈ V)
16 ssun2 3739 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
17 ssexg 4732 . . . . . 6 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1816, 13, 17sylancr 694 . . . . 5 (𝑅 ∈ TosetRel → 𝐵 ∈ V)
19 elfiun 8219 . . . . 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 691 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛))))
213, 4ordtbaslem 20802 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
2221, 1syl6eqss 3618 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ (𝐴𝐵))
23 ssun1 3738 . . . . . . 7 (𝐴𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶)
2422, 23syl6ss 3580 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ ((𝐴𝐵) ∪ 𝐶))
2524sseld 3567 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
26 cnvtsr 17045 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
27 df-rn 5049 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
28 eqid 2610 . . . . . . . . . . 11 ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
2927, 28ordtbaslem 20802 . . . . . . . . . 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 17044 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
323psrn 17032 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
3331, 32syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅)
34 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
35 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
3634, 35brcnv 5227 . . . . . . . . . . . . . . . . 17 (𝑦𝑅𝑥𝑥𝑅𝑦)
3736bicomi 213 . . . . . . . . . . . . . . . 16 (𝑥𝑅𝑦𝑦𝑅𝑥)
3837notbii 309 . . . . . . . . . . . . . . 15 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥)
3938a1i 11 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
4033, 39rabeqbidv 3168 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
4133, 40mpteq12dv 4663 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4241rneqd 5274 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
435, 42syl5eq 2656 . . . . . . . . . 10 (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
4443fveq2d 6107 . . . . . . . . 9 (𝑅 ∈ TosetRel → (fi‘𝐵) = (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})))
4530, 44, 433eqtr4d 2654 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘𝐵) = 𝐵)
4645, 16syl6eqss 3618 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ (𝐴𝐵))
4746, 23syl6ss 3580 . . . . . 6 (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ ((𝐴𝐵) ∪ 𝐶))
4847sseld 3567 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
49 ssun2 3739 . . . . . . . 8 𝐶 ⊆ ((𝐴𝐵) ∪ 𝐶)
5021, 4syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐴) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
5150eleq2d 2673 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})))
52 vex 3176 . . . . . . . . . . . . . . 15 𝑚 ∈ V
53 breq2 4587 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑦𝑅𝑥𝑦𝑅𝑎))
5453notbid 307 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
5554rabbidv 3164 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5655cbvmptv 4678 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
5756elrnmpt 5293 . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
6045, 5syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑅 ∈ TosetRel → (fi‘𝐵) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
6160eleq2d 2673 . . . . . . . . . . . . . 14 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))
62 vex 3176 . . . . . . . . . . . . . . 15 𝑛 ∈ V
63 breq1 4586 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (𝑥𝑅𝑦𝑏𝑅𝑦))
6463notbid 307 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦))
6564rabbidv 3164 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑏 → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6665cbvmptv 4678 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
6766elrnmpt 5293 . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
7059, 69anbi12d 743 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})))
71 reeanv 3086 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
72 ineq12 3771 . . . . . . . . . . . . . . . 16 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
73 inrab 3858 . . . . . . . . . . . . . . . 16 ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}
7472, 73syl6eq 2660 . . . . . . . . . . . . . . 15 ((𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7574reximi 2994 . . . . . . . . . . . . . 14 (∃𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7675reximi 2994 . . . . . . . . . . . . 13 (∃𝑎𝑋𝑏𝑋 (𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7771, 76sylbir 224 . . . . . . . . . . . 12 ((∃𝑎𝑋 𝑚 = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏𝑋 𝑛 = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
7870, 77syl6bi 242 . . . . . . . . . . 11 (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
7978imp 444 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎𝑋𝑏𝑋 (𝑚𝑛) = {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8052inex1 4727 . . . . . . . . . . 11 (𝑚𝑛) ∈ V
81 eqid 2610 . . . . . . . . . . . 12 (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
8281elrnmpt2g 6670 . . . . . . . . . . 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 223 . . . . . . . . 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 2699 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ 𝐶)
8749, 86sseldi 3566 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶))
88 eleq1 2676 . . . . . . 7 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑚𝑛) ∈ ((𝐴𝐵) ∪ 𝐶)))
8987, 88syl5ibrcom 236 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9089rexlimdvva 3020 . . . . 5 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9125, 48, 903jaod 1384 . . . 4 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚𝑛)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9220, 91sylbid 229 . . 3 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ((𝐴𝐵) ∪ 𝐶)))
9392ssrdv 3574 . 2 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ((𝐴𝐵) ∪ 𝐶))
94 ssfii 8208 . . . 4 ((𝐴𝐵) ∈ V → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9513, 94syl 17 . . 3 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
9695adantr 480 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → (𝐴𝐵) ⊆ (fi‘(𝐴𝐵)))
97 simprl 790 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
98 eqidd 2611 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎})
9955eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}))
10099rspcev 3282 . . . . . . . . . . . . . 14 ((𝑎𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
10197, 98, 100syl2anc 691 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1028adantr 480 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑋 ∈ V)
103 rabexg 4739 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
104 eqid 2610 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
105104elrnmpt 5293 . . . . . . . . . . . . . 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 246 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
108107, 4syl6eleqr 2699 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴)
1091, 108sseldi 3566 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴𝐵))
11096, 109sseldd 3569 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)))
111 simprr 792 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
112 eqidd 2611 . . . . . . . . . . . . . 14 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦})
11365eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑥 = 𝑏 → ({𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ↔ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}))
114113rspcev 3282 . . . . . . . . . . . . . 14 ((𝑏𝑋 ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
115111, 112, 114syl2anc 691 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ∃𝑥𝑋 {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
116 rabexg 4739 . . . . . . . . . . . . . 14 (𝑋 ∈ V → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V)
117 eqid 2610 . . . . . . . . . . . . . . 15 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
118117elrnmpt 5293 . . . . . . . . . . . . . 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 246 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
121120, 5syl6eleqr 2699 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵)
12216, 121sseldi 3566 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴𝐵))
12396, 122sseldd 3569 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵)))
124 fiin 8211 . . . . . . . . 9 (({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴𝐵)) ∧ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴𝐵))) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
125110, 123, 124syl2anc 691 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴𝐵)))
12673, 125syl5eqelr 2693 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑎𝑋𝑏𝑋)) → {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
127126ralrimivva 2954 . . . . . 6 (𝑅 ∈ TosetRel → ∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)))
12881fmpt2 7126 . . . . . 6 (∀𝑎𝑋𝑏𝑋 {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴𝐵)) ↔ (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
129127, 128sylib 207 . . . . 5 (𝑅 ∈ TosetRel → (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴𝐵)))
130 frn 5966 . . . . 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 3612 . . 3 (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴𝐵)))
13395, 132unssd 3751 . 2 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴𝐵)))
13493, 133eqssd 3585 1 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3o 1030   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  cun 3538  cin 3539  wss 3540  {csn 4125   cuni 4372   class class class wbr 4583  cmpt 4643   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  wf 5800  cfv 5804  cmpt2 6551  ficfi 8199  PosetRelcps 17021   TosetRel ctsr 17022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-ps 17023  df-tsr 17024
This theorem is referenced by:  ordtbas  20806  leordtval  20827
  Copyright terms: Public domain W3C validator