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Theorem ordtbaslem 22555
Description: Lemma for ordtbas 22559. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypotheses
Ref Expression
ordtval.1 𝑋 = dom 𝑅
ordtval.2 𝐴 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
Assertion
Ref Expression
ordtbaslem (𝑅 ∈ TosetRel β†’ (fiβ€˜π΄) = 𝐴)
Distinct variable groups:   π‘₯,𝑦,𝑅   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)

Proof of Theorem ordtbaslem
Dummy variables π‘Ž 𝑏 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3anrot 1101 . . . . . . . . . . . . 13 ((𝑦 ∈ 𝑋 ∧ π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
2 ordtval.1 . . . . . . . . . . . . . 14 𝑋 = dom 𝑅
32tsrlemax 18482 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑦 ∈ 𝑋 ∧ π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ (𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏)))
41, 3sylan2br 596 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏)))
543exp2 1355 . . . . . . . . . . 11 (𝑅 ∈ TosetRel β†’ (π‘Ž ∈ 𝑋 β†’ (𝑏 ∈ 𝑋 β†’ (𝑦 ∈ 𝑋 β†’ (𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏))))))
65imp42 428 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) β†’ (𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏)))
76notbid 318 . . . . . . . . 9 (((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) β†’ (Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ Β¬ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏)))
8 ioran 983 . . . . . . . . 9 (Β¬ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏) ↔ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏))
97, 8bitrdi 287 . . . . . . . 8 (((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) β†’ (Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)))
109rabbidva 3417 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)})
11 ifcl 4536 . . . . . . . . 9 ((𝑏 ∈ 𝑋 ∧ π‘Ž ∈ 𝑋) β†’ if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ∈ 𝑋)
1211ancoms 460 . . . . . . . 8 ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ∈ 𝑋)
13 dmexg 7845 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel β†’ dom 𝑅 ∈ V)
142, 13eqeltrid 2842 . . . . . . . . . . 11 (𝑅 ∈ TosetRel β†’ 𝑋 ∈ V)
1514adantr 482 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ 𝑋 ∈ V)
16 rabexg 5293 . . . . . . . . . 10 (𝑋 ∈ V β†’ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ V)
1715, 16syl 17 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ V)
1810, 17eqeltrd 2838 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ V)
19 eqid 2737 . . . . . . . . . 10 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
20 breq2 5114 . . . . . . . . . . . 12 (π‘₯ = if(π‘Žπ‘…π‘, 𝑏, π‘Ž) β†’ (𝑦𝑅π‘₯ ↔ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)))
2120notbid 318 . . . . . . . . . . 11 (π‘₯ = if(π‘Žπ‘…π‘, 𝑏, π‘Ž) β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)))
2221rabbidv 3418 . . . . . . . . . 10 (π‘₯ = if(π‘Žπ‘…π‘, 𝑏, π‘Ž) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)})
2319, 22elrnmpt1s 5917 . . . . . . . . 9 ((if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ V) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
24 ordtval.2 . . . . . . . . 9 𝐴 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
2523, 24eleqtrrdi 2849 . . . . . . . 8 ((if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ V) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ 𝐴)
2612, 18, 25syl2an2 685 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ 𝐴)
2710, 26eqeltrrd 2839 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴)
2827ralrimivva 3198 . . . . 5 (𝑅 ∈ TosetRel β†’ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴)
29 rabexg 5293 . . . . . . . 8 (𝑋 ∈ V β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ V)
3014, 29syl 17 . . . . . . 7 (𝑅 ∈ TosetRel β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ V)
3130ralrimivw 3148 . . . . . 6 (𝑅 ∈ TosetRel β†’ βˆ€π‘Ž ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ V)
32 breq2 5114 . . . . . . . . . 10 (π‘₯ = π‘Ž β†’ (𝑦𝑅π‘₯ ↔ π‘¦π‘…π‘Ž))
3332notbid 318 . . . . . . . . 9 (π‘₯ = π‘Ž β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ π‘¦π‘…π‘Ž))
3433rabbidv 3418 . . . . . . . 8 (π‘₯ = π‘Ž β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
3534cbvmptv 5223 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (π‘Ž ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
36 ineq1 4170 . . . . . . . . . 10 (𝑧 = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} β†’ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) = ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}))
37 inrab 4271 . . . . . . . . . 10 ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)}
3836, 37eqtrdi 2793 . . . . . . . . 9 (𝑧 = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} β†’ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)})
3938eleq1d 2823 . . . . . . . 8 (𝑧 = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} β†’ ((𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4039ralbidv 3175 . . . . . . 7 (𝑧 = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} β†’ (βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4135, 40ralrnmptw 7049 . . . . . 6 (βˆ€π‘Ž ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ V β†’ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4231, 41syl 17 . . . . 5 (𝑅 ∈ TosetRel β†’ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴))
4328, 42mpbird 257 . . . 4 (𝑅 ∈ TosetRel β†’ βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴)
44 rabexg 5293 . . . . . . . 8 (𝑋 ∈ V β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} ∈ V)
4514, 44syl 17 . . . . . . 7 (𝑅 ∈ TosetRel β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} ∈ V)
4645ralrimivw 3148 . . . . . 6 (𝑅 ∈ TosetRel β†’ βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} ∈ V)
47 breq2 5114 . . . . . . . . . 10 (π‘₯ = 𝑏 β†’ (𝑦𝑅π‘₯ ↔ 𝑦𝑅𝑏))
4847notbid 318 . . . . . . . . 9 (π‘₯ = 𝑏 β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝑦𝑅𝑏))
4948rabbidv 3418 . . . . . . . 8 (π‘₯ = 𝑏 β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏})
5049cbvmptv 5223 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏})
51 ineq2 4171 . . . . . . . 8 (𝑀 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} β†’ (𝑧 ∩ 𝑀) = (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}))
5251eleq1d 2823 . . . . . . 7 (𝑀 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} β†’ ((𝑧 ∩ 𝑀) ∈ 𝐴 ↔ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5350, 52ralrnmptw 7049 . . . . . 6 (βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} ∈ V β†’ (βˆ€π‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})(𝑧 ∩ 𝑀) ∈ 𝐴 ↔ βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5446, 53syl 17 . . . . 5 (𝑅 ∈ TosetRel β†’ (βˆ€π‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})(𝑧 ∩ 𝑀) ∈ 𝐴 ↔ βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5554ralbidv 3175 . . . 4 (𝑅 ∈ TosetRel β†’ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})(𝑧 ∩ 𝑀) ∈ 𝐴 ↔ βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5643, 55mpbird 257 . . 3 (𝑅 ∈ TosetRel β†’ βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})(𝑧 ∩ 𝑀) ∈ 𝐴)
5724raleqi 3314 . . . 4 (βˆ€π‘€ ∈ 𝐴 (𝑧 ∩ 𝑀) ∈ 𝐴 ↔ βˆ€π‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})(𝑧 ∩ 𝑀) ∈ 𝐴)
5824, 57raleqbii 3318 . . 3 (βˆ€π‘§ ∈ 𝐴 βˆ€π‘€ ∈ 𝐴 (𝑧 ∩ 𝑀) ∈ 𝐴 ↔ βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})(𝑧 ∩ 𝑀) ∈ 𝐴)
5956, 58sylibr 233 . 2 (𝑅 ∈ TosetRel β†’ βˆ€π‘§ ∈ 𝐴 βˆ€π‘€ ∈ 𝐴 (𝑧 ∩ 𝑀) ∈ 𝐴)
6014pwexd 5339 . . . 4 (𝑅 ∈ TosetRel β†’ 𝒫 𝑋 ∈ V)
61 ssrab2 4042 . . . . . . . 8 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋
6214adantr 482 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 ∈ V)
63 elpw2g 5306 . . . . . . . . 9 (𝑋 ∈ V β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋))
6462, 63syl 17 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋))
6561, 64mpbiri 258 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋)
6665fmpttd 7068 . . . . . 6 (𝑅 ∈ TosetRel β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}):π‘‹βŸΆπ’« 𝑋)
6766frnd 6681 . . . . 5 (𝑅 ∈ TosetRel β†’ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) βŠ† 𝒫 𝑋)
6824, 67eqsstrid 3997 . . . 4 (𝑅 ∈ TosetRel β†’ 𝐴 βŠ† 𝒫 𝑋)
6960, 68ssexd 5286 . . 3 (𝑅 ∈ TosetRel β†’ 𝐴 ∈ V)
70 inficl 9368 . . 3 (𝐴 ∈ V β†’ (βˆ€π‘§ ∈ 𝐴 βˆ€π‘€ ∈ 𝐴 (𝑧 ∩ 𝑀) ∈ 𝐴 ↔ (fiβ€˜π΄) = 𝐴))
7169, 70syl 17 . 2 (𝑅 ∈ TosetRel β†’ (βˆ€π‘§ ∈ 𝐴 βˆ€π‘€ ∈ 𝐴 (𝑧 ∩ 𝑀) ∈ 𝐴 ↔ (fiβ€˜π΄) = 𝐴))
7259, 71mpbid 231 1 (𝑅 ∈ TosetRel β†’ (fiβ€˜π΄) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  ifcif 4491  π’« cpw 4565   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5638  ran crn 5639  β€˜cfv 6501  ficfi 9353   TosetRel ctsr 18461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-er 8655  df-en 8891  df-fin 8894  df-fi 9354  df-ps 18462  df-tsr 18463
This theorem is referenced by:  ordtbas2  22558
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