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Theorem dfrtrcl2 15013
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 2731 . . . . . 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 5902 . . . . . . . . . . . . 13 (π‘₯ = 𝑅 β†’ dom π‘₯ = dom 𝑅)
3 rneq 5934 . . . . . . . . . . . . 13 (π‘₯ = 𝑅 β†’ ran π‘₯ = ran 𝑅)
42, 3uneq12d 4163 . . . . . . . . . . . 12 (π‘₯ = 𝑅 β†’ (dom π‘₯ βˆͺ ran π‘₯) = (dom 𝑅 βˆͺ ran 𝑅))
54reseq2d 5980 . . . . . . . . . . 11 (π‘₯ = 𝑅 β†’ ( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) = ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)))
65sseq1d 4012 . . . . . . . . . 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 4012 . . . . . . . . . 10 (π‘₯ = 𝑅 β†’ (π‘₯ βŠ† 𝑧 ↔ 𝑅 βŠ† 𝑧))
96, 83anbi12d 1435 . . . . . . . . 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 2799 . . . . . . . 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 4954 . . . . . . 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 480 . . . . . 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 483 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ 𝑅 ∈ V)
14 dfrtrcl2.1 . . . . . . . . . . . . 13 (πœ‘ β†’ Rel 𝑅)
15 relfld 6273 . . . . . . . . . . . . 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 2736 . . . . . . . . . . 11 (πœ‘ β†’ (dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅)
1817adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ 𝑅 ∈ V) β†’ (dom 𝑅 βˆͺ ran 𝑅) = βˆͺ βˆͺ 𝑅)
1914adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑅 ∈ V) β†’ Rel 𝑅)
2019, 13rtrclreclem2 15010 . . . . . . . . . . 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 5980 . . . . . . . . . . . 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 4012 . . . . . . . . . . 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 15008 . . . . . . . . 9 ((πœ‘ ∧ 𝑅 ∈ V) β†’ 𝑅 βŠ† (t*recβ€˜π‘…))
2714rtrclreclem3 15011 . . . . . . . . . 10 (πœ‘ β†’ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))
2827adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))
29 fvex 6903 . . . . . . . . . . 11 (t*recβ€˜π‘…) ∈ V
30 sseq2 4007 . . . . . . . . . . . . . 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 4007 . . . . . . . . . . . . . 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 5863 . . . . . . . . . . . . . . 15 (𝑧 = (t*recβ€˜π‘…) β†’ (𝑧 ∘ 𝑧) = ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)))
3433, 32sseq12d 4014 . . . . . . . . . . . . . 14 (𝑧 = (t*recβ€˜π‘…) β†’ ((𝑧 ∘ 𝑧) βŠ† 𝑧 ↔ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…)))
3530, 31, 343anbi123d 1434 . . . . . . . . . . . . 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 1928 . . . . . . . . . . 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 3661 . . . . . . . . . . 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 585 . . . . . . . . . 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 479 . . . . . . . . 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 1340 . . . . . . . 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 4334 . . . . . . 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 5336 . . . . . . 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 7004 . . . . 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 4966 . . . . . . 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 3476 . . . . . . . . . . 11 𝑠 ∈ V
49 sseq2 4007 . . . . . . . . . . . 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 4007 . . . . . . . . . . . 12 (𝑧 = 𝑠 β†’ (𝑅 βŠ† 𝑧 ↔ 𝑅 βŠ† 𝑠))
51 id 22 . . . . . . . . . . . . . 14 (𝑧 = 𝑠 β†’ 𝑧 = 𝑠)
5251, 51coeq12d 5863 . . . . . . . . . . . . 13 (𝑧 = 𝑠 β†’ (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
5352, 51sseq12d 4014 . . . . . . . . . . . 12 (𝑧 = 𝑠 β†’ ((𝑧 ∘ 𝑧) βŠ† 𝑧 ↔ (𝑠 ∘ 𝑠) βŠ† 𝑠))
5449, 50, 533anbi123d 1434 . . . . . . . . . . 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 3667 . . . . . . . . . 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 15012 . . . . . . . . . . 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 2180 . . . . . . . . . 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 3143 . . . . . . . 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 4967 . . . . . . . 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 479 . . . . . 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 3998 . . . . 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 2770 . . . 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 14939 . . . . 5 t* = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
66 fveq1 6889 . . . . . . 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 2732 . . . . . 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 339 . . . . 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 411 . 2 (πœ‘ β†’ (𝑅 ∈ V β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…)))
72 fvprc 6882 . . 3 (Β¬ 𝑅 ∈ V β†’ (t*β€˜π‘…) = βˆ…)
73 fvprc 6882 . . . 4 (Β¬ 𝑅 ∈ V β†’ (t*recβ€˜π‘…) = βˆ…)
7473eqcomd 2736 . . 3 (Β¬ 𝑅 ∈ V β†’ βˆ… = (t*recβ€˜π‘…))
7572, 74eqtrd 2770 . 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 394   ∧ w3a 1085  βˆ€wal 1537   = wceq 1539   ∈ wcel 2104  {cab 2707   β‰  wne 2938  βˆ€wral 3059  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  βˆ…c0 4321  βˆͺ cuni 4907  βˆ© cint 4949   ↦ cmpt 5230   I cid 5572  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679  Rel wrel 5680  β€˜cfv 6542  t*crtcl 14937  t*reccrtrcl 15006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-rtrcl 14939  df-relexp 14971  df-rtrclrec 15007
This theorem is referenced by:  rtrclind  15016
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