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Theorem dfrtrcl2 15009
Description: The two definitions t* and t*rec of the reflexive, transitive closure coincide if 𝑅 is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
Hypothesis
Ref Expression
dfrtrcl2.1 (πœ‘ β†’ Rel 𝑅)
Assertion
Ref Expression
dfrtrcl2 (πœ‘ β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…))
Proof of Theorem dfrtrcl2
Dummy variables π‘₯ 𝑧 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2734 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)}) = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)}))
2 dmeq 5904 . . . . . . . . . . . . 13 (π‘₯ = 𝑅 β†’ dom π‘₯ = dom 𝑅)
3 rneq 5936 . . . . . . . . . . . . 13 (π‘₯ = 𝑅 β†’ ran π‘₯ = ran 𝑅)
42, 3uneq12d 4165 . . . . . . . . . . . 12 (π‘₯ = 𝑅 β†’ (dom π‘₯ βˆͺ ran π‘₯) = (dom 𝑅 βˆͺ ran 𝑅))
54reseq2d 5982 . . . . . . . . . . 11 (π‘₯ = 𝑅 β†’ ( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) = ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)))
65sseq1d 4014 . . . . . . . . . 10 (π‘₯ = 𝑅 β†’ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ↔ ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧))
7 id 22 . . . . . . . . . . 11 (π‘₯ = 𝑅 β†’ π‘₯ = 𝑅)
87sseq1d 4014 . . . . . . . . . 10 (π‘₯ = 𝑅 β†’ (π‘₯ βŠ† 𝑧 ↔ 𝑅 βŠ† 𝑧))
96, 83anbi12d 1438 . . . . . . . . 9 (π‘₯ = 𝑅 β†’ ((( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧) ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)))
109abbidv 2802 . . . . . . . 8 (π‘₯ = 𝑅 β†’ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} = {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
1110inteqd 4956 . . . . . . 7 (π‘₯ = 𝑅 β†’ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} = ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
1211adantl 483 . . . . . 6 (((πœ‘ ∧ 𝑅 ∈ V) ∧ π‘₯ = 𝑅) β†’ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} = ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
13 simpr 486 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ 𝑅 ∈ V)
14 dfrtrcl2.1 . . . . . . . . . . . . 13 (πœ‘ β†’ Rel 𝑅)
15 relfld 6275 . . . . . . . . . . . . 13 (Rel 𝑅 β†’ βˆͺ βˆͺ 𝑅 = (dom 𝑅 βˆͺ ran 𝑅))
1614, 15syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ βˆͺ 𝑅 = (dom 𝑅 βˆͺ ran 𝑅))
1716eqcomd 2739 . . . . . . . . . . 11 (πœ‘ β†’ (dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅)
1817adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑅 ∈ V) β†’ (dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅)
1914adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑅 ∈ V) β†’ Rel 𝑅)
2019, 13rtrclreclem2 15006 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ( I β†Ύ βˆͺ βˆͺ 𝑅) βŠ† (t*recβ€˜π‘…))
21 id 22 . . . . . . . . . . . . 13 ((dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅 β†’ (dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅)
2221reseq2d 5982 . . . . . . . . . . . 12 ((dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅 β†’ ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) = ( I β†Ύ βˆͺ βˆͺ 𝑅))
2322sseq1d 4014 . . . . . . . . . . 11 ((dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅 β†’ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ↔ ( I β†Ύ βˆͺ βˆͺ 𝑅) βŠ† (t*recβ€˜π‘…)))
2420, 23imbitrrid 245 . . . . . . . . . 10 ((dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅 β†’ ((πœ‘ ∧ 𝑅 ∈ V) β†’ ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…)))
2518, 24mpcom 38 . . . . . . . . 9 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…))
2613rtrclreclem1 15004 . . . . . . . . 9 ((πœ‘ ∧ 𝑅 ∈ V) β†’ 𝑅 βŠ† (t*recβ€˜π‘…))
2714rtrclreclem3 15007 . . . . . . . . . 10 (πœ‘ β†’ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))
2827adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))
29 fvex 6905 . . . . . . . . . . 11 (t*recβ€˜π‘…) ∈ V
30 sseq2 4009 . . . . . . . . . . . . . 14 (𝑧 = (t*recβ€˜π‘…) β†’ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ↔ ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…)))
31 sseq2 4009 . . . . . . . . . . . . . 14 (𝑧 = (t*recβ€˜π‘…) β†’ (𝑅 βŠ† 𝑧 ↔ 𝑅 βŠ† (t*recβ€˜π‘…)))
32 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = (t*recβ€˜π‘…) β†’ 𝑧 = (t*recβ€˜π‘…))
3332, 32coeq12d 5865 . . . . . . . . . . . . . . 15 (𝑧 = (t*recβ€˜π‘…) β†’ (𝑧 ∘ 𝑧) = ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)))
3433, 32sseq12d 4016 . . . . . . . . . . . . . 14 (𝑧 = (t*recβ€˜π‘…) β†’ ((𝑧 ∘ 𝑧) βŠ† 𝑧 ↔ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…)))
3530, 31, 343anbi123d 1437 . . . . . . . . . . . . 13 (𝑧 = (t*recβ€˜π‘…) β†’ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧) ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ∧ 𝑅 βŠ† (t*recβ€˜π‘…) ∧ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))))
3635a1i 11 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑧 = (t*recβ€˜π‘…) β†’ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧) ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ∧ 𝑅 βŠ† (t*recβ€˜π‘…) ∧ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…)))))
3736alrimiv 1931 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘§(𝑧 = (t*recβ€˜π‘…) β†’ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧) ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ∧ 𝑅 βŠ† (t*recβ€˜π‘…) ∧ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…)))))
38 elabgt 3663 . . . . . . . . . . 11 (((t*recβ€˜π‘…) ∈ V ∧ βˆ€π‘§(𝑧 = (t*recβ€˜π‘…) β†’ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧) ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ∧ 𝑅 βŠ† (t*recβ€˜π‘…) ∧ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))))) β†’ ((t*recβ€˜π‘…) ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ∧ 𝑅 βŠ† (t*recβ€˜π‘…) ∧ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))))
3929, 37, 38sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ ((t*recβ€˜π‘…) ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ∧ 𝑅 βŠ† (t*recβ€˜π‘…) ∧ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))))
4039adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((t*recβ€˜π‘…) ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† (t*recβ€˜π‘…) ∧ 𝑅 βŠ† (t*recβ€˜π‘…) ∧ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))))
4125, 26, 28, 40mpbir3and 1343 . . . . . . . 8 ((πœ‘ ∧ 𝑅 ∈ V) β†’ (t*recβ€˜π‘…) ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
4241ne0d 4336 . . . . . . 7 ((πœ‘ ∧ 𝑅 ∈ V) β†’ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} β‰  βˆ…)
43 intex 5338 . . . . . . 7 ({𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} β‰  βˆ… ↔ ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} ∈ V)
4442, 43sylib 217 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} ∈ V)
451, 12, 13, 44fvmptd 7006 . . . . 5 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})β€˜π‘…) = ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
46 intss1 4968 . . . . . . 7 ((t*recβ€˜π‘…) ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} β†’ ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} βŠ† (t*recβ€˜π‘…))
4741, 46syl 17 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} βŠ† (t*recβ€˜π‘…))
48 vex 3479 . . . . . . . . . . 11 𝑠 ∈ V
49 sseq2 4009 . . . . . . . . . . . 12 (𝑧 = 𝑠 β†’ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ↔ ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑠))
50 sseq2 4009 . . . . . . . . . . . 12 (𝑧 = 𝑠 β†’ (𝑅 βŠ† 𝑧 ↔ 𝑅 βŠ† 𝑠))
51 id 22 . . . . . . . . . . . . . 14 (𝑧 = 𝑠 β†’ 𝑧 = 𝑠)
5251, 51coeq12d 5865 . . . . . . . . . . . . 13 (𝑧 = 𝑠 β†’ (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
5352, 51sseq12d 4016 . . . . . . . . . . . 12 (𝑧 = 𝑠 β†’ ((𝑧 ∘ 𝑧) βŠ† 𝑧 ↔ (𝑠 ∘ 𝑠) βŠ† 𝑠))
5449, 50, 533anbi123d 1437 . . . . . . . . . . 11 (𝑧 = 𝑠 β†’ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧) ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑠 ∧ 𝑅 βŠ† 𝑠 ∧ (𝑠 ∘ 𝑠) βŠ† 𝑠)))
5548, 54elab 3669 . . . . . . . . . 10 (𝑠 ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} ↔ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑠 ∧ 𝑅 βŠ† 𝑠 ∧ (𝑠 ∘ 𝑠) βŠ† 𝑠))
5614rtrclreclem4 15008 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑠 ∧ 𝑅 βŠ† 𝑠 ∧ (𝑠 ∘ 𝑠) βŠ† 𝑠) β†’ (t*recβ€˜π‘…) βŠ† 𝑠))
575619.21bi 2183 . . . . . . . . . 10 (πœ‘ β†’ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑠 ∧ 𝑅 βŠ† 𝑠 ∧ (𝑠 ∘ 𝑠) βŠ† 𝑠) β†’ (t*recβ€˜π‘…) βŠ† 𝑠))
5855, 57biimtrid 241 . . . . . . . . 9 (πœ‘ β†’ (𝑠 ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} β†’ (t*recβ€˜π‘…) βŠ† 𝑠))
5958ralrimiv 3146 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘  ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} (t*recβ€˜π‘…) βŠ† 𝑠)
60 ssint 4969 . . . . . . . 8 ((t*recβ€˜π‘…) βŠ† ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} ↔ βˆ€π‘  ∈ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} (t*recβ€˜π‘…) βŠ† 𝑠)
6159, 60sylibr 233 . . . . . . 7 (πœ‘ β†’ (t*recβ€˜π‘…) βŠ† ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
6261adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ (t*recβ€˜π‘…) βŠ† ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
6347, 62eqssd 4000 . . . . 5 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ∩ {𝑧 ∣ (( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑧 ∧ 𝑅 βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)} = (t*recβ€˜π‘…))
6445, 63eqtrd 2773 . . . 4 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})β€˜π‘…) = (t*recβ€˜π‘…))
65 df-rtrcl 14935 . . . . 5 t* = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
66 fveq1 6891 . . . . . . 7 (t* = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)}) β†’ (t*β€˜π‘…) = ((π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})β€˜π‘…))
6766eqeq1d 2735 . . . . . 6 (t* = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)}) β†’ ((t*β€˜π‘…) = (t*recβ€˜π‘…) ↔ ((π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})β€˜π‘…) = (t*recβ€˜π‘…)))
6867imbi2d 341 . . . . 5 (t* = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)}) β†’ (((πœ‘ ∧ 𝑅 ∈ V) β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…)) ↔ ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})β€˜π‘…) = (t*recβ€˜π‘…))))
6965, 68ax-mp 5 . . . 4 (((πœ‘ ∧ 𝑅 ∈ V) β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…)) ↔ ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})β€˜π‘…) = (t*recβ€˜π‘…)))
7064, 69mpbir 230 . . 3 ((πœ‘ ∧ 𝑅 ∈ V) β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…))
7170ex 414 . 2 (πœ‘ β†’ (𝑅 ∈ V β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…)))
72 fvprc 6884 . . 3 (Β¬ 𝑅 ∈ V β†’ (t*β€˜π‘…) = βˆ…)
73 fvprc 6884 . . . 4 (Β¬ 𝑅 ∈ V β†’ (t*recβ€˜π‘…) = βˆ…)
7473eqcomd 2739 . . 3 (Β¬ 𝑅 ∈ V β†’ βˆ… = (t*recβ€˜π‘…))
7572, 74eqtrd 2773 . 2 (Β¬ 𝑅 ∈ V β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…))
7671, 75pm2.61d1 180 1 (πœ‘ β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909  βˆ© cint 4951   ↦ cmpt 5232   I cid 5574  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   ∘ ccom 5681  Rel wrel 5682  β€˜cfv 6544  t*crtcl 14933  t*reccrtrcl 15002
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-rtrcl 14935  df-relexp 14967  df-rtrclrec 15003
This theorem is referenced by:  rtrclind  15012
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