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Theorem ordtbaslem 22699
Description: Lemma for ordtbas 22703. 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 1100 . . . . . . . . . . . . 13 ((𝑦 ∈ 𝑋 ∧ π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
2 ordtval.1 . . . . . . . . . . . . . 14 𝑋 = dom 𝑅
32tsrlemax 18541 . . . . . . . . . . . . 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 595 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏)))
543exp2 1354 . . . . . . . . . . 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 427 . . . . . . . . . 10 (((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) β†’ (𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ↔ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏)))
76notbid 317 . . . . . . . . 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 982 . . . . . . . . 9 (Β¬ (π‘¦π‘…π‘Ž ∨ 𝑦𝑅𝑏) ↔ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏))
97, 8bitrdi 286 . . . . . . . 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 3439 . . . . . . 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 4573 . . . . . . . . 9 ((𝑏 ∈ 𝑋 ∧ π‘Ž ∈ 𝑋) β†’ if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ∈ 𝑋)
1211ancoms 459 . . . . . . . 8 ((π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) β†’ if(π‘Žπ‘…π‘, 𝑏, π‘Ž) ∈ 𝑋)
13 dmexg 7896 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel β†’ dom 𝑅 ∈ V)
142, 13eqeltrid 2837 . . . . . . . . . . 11 (𝑅 ∈ TosetRel β†’ 𝑋 ∈ V)
1514adantr 481 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ 𝑋 ∈ V)
16 rabexg 5331 . . . . . . . . . 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 2833 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ V)
19 eqid 2732 . . . . . . . . . 10 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
20 breq2 5152 . . . . . . . . . . . 12 (π‘₯ = if(π‘Žπ‘…π‘, 𝑏, π‘Ž) β†’ (𝑦𝑅π‘₯ ↔ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)))
2120notbid 317 . . . . . . . . . . 11 (π‘₯ = if(π‘Žπ‘…π‘, 𝑏, π‘Ž) β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)))
2221rabbidv 3440 . . . . . . . . . 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 5956 . . . . . . . . 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 2844 . . . . . . . 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 684 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅if(π‘Žπ‘…π‘, 𝑏, π‘Ž)} ∈ 𝐴)
2710, 26eqeltrrd 2834 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (π‘Ž ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) β†’ {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴)
2827ralrimivva 3200 . . . . 5 (𝑅 ∈ TosetRel β†’ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)} ∈ 𝐴)
29 rabexg 5331 . . . . . . . 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 3150 . . . . . 6 (𝑅 ∈ TosetRel β†’ βˆ€π‘Ž ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∈ V)
32 breq2 5152 . . . . . . . . . 10 (π‘₯ = π‘Ž β†’ (𝑦𝑅π‘₯ ↔ π‘¦π‘…π‘Ž))
3332notbid 317 . . . . . . . . 9 (π‘₯ = π‘Ž β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ π‘¦π‘…π‘Ž))
3433rabbidv 3440 . . . . . . . 8 (π‘₯ = π‘Ž β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
3534cbvmptv 5261 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (π‘Ž ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž})
36 ineq1 4205 . . . . . . . . . 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 4306 . . . . . . . . . 10 ({𝑦 ∈ 𝑋 ∣ Β¬ π‘¦π‘…π‘Ž} ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (Β¬ π‘¦π‘…π‘Ž ∧ Β¬ 𝑦𝑅𝑏)}
3836, 37eqtrdi 2788 . . . . . . . . 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 2818 . . . . . . . 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 3177 . . . . . . 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 7095 . . . . . 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 256 . . . 4 (𝑅 ∈ TosetRel β†’ βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})βˆ€π‘ ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴)
44 rabexg 5331 . . . . . . . 8 (𝑋 ∈ V β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} ∈ V)
4514, 44syl 17 . . . . . . 7 (𝑅 ∈ TosetRel β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} ∈ V)
4645ralrimivw 3150 . . . . . 6 (𝑅 ∈ TosetRel β†’ βˆ€π‘ ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} ∈ V)
47 breq2 5152 . . . . . . . . . 10 (π‘₯ = 𝑏 β†’ (𝑦𝑅π‘₯ ↔ 𝑦𝑅𝑏))
4847notbid 317 . . . . . . . . 9 (π‘₯ = 𝑏 β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝑦𝑅𝑏))
4948rabbidv 3440 . . . . . . . 8 (π‘₯ = 𝑏 β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏})
5049cbvmptv 5261 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏})
51 ineq2 4206 . . . . . . . 8 (𝑀 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} β†’ (𝑧 ∩ 𝑀) = (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}))
5251eleq1d 2818 . . . . . . 7 (𝑀 = {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏} β†’ ((𝑧 ∩ 𝑀) ∈ 𝐴 ↔ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅𝑏}) ∈ 𝐴))
5350, 52ralrnmptw 7095 . . . . . 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 3177 . . . 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 256 . . 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 3323 . . . 4 (βˆ€π‘€ ∈ 𝐴 (𝑧 ∩ 𝑀) ∈ 𝐴 ↔ βˆ€π‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})(𝑧 ∩ 𝑀) ∈ 𝐴)
5824, 57raleqbii 3338 . . 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 5377 . . . 4 (𝑅 ∈ TosetRel β†’ 𝒫 𝑋 ∈ V)
61 ssrab2 4077 . . . . . . . 8 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋
6214adantr 481 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 ∈ V)
63 elpw2g 5344 . . . . . . . . 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 257 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋)
6665fmpttd 7116 . . . . . 6 (𝑅 ∈ TosetRel β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}):π‘‹βŸΆπ’« 𝑋)
6766frnd 6725 . . . . 5 (𝑅 ∈ TosetRel β†’ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) βŠ† 𝒫 𝑋)
6824, 67eqsstrid 4030 . . . 4 (𝑅 ∈ TosetRel β†’ 𝐴 βŠ† 𝒫 𝑋)
6960, 68ssexd 5324 . . 3 (𝑅 ∈ TosetRel β†’ 𝐴 ∈ V)
70 inficl 9422 . . 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  ifcif 4528  π’« cpw 4602   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677  β€˜cfv 6543  ficfi 9407   TosetRel ctsr 18520
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-ps 18521  df-tsr 18522
This theorem is referenced by:  ordtbas2  22702
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