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Theorem dfrtrcl2 15006
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 5902 . . . . . . . . . . . . 13 (π‘₯ = 𝑅 β†’ dom π‘₯ = dom 𝑅)
3 rneq 5934 . . . . . . . . . . . . 13 (π‘₯ = 𝑅 β†’ ran π‘₯ = ran 𝑅)
42, 3uneq12d 4164 . . . . . . . . . . . 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 4013 . . . . . . . . . 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 4013 . . . . . . . . . 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 4955 . . . . . . 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 6272 . . . . . . . . . . . . 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 15003 . . . . . . . . . . 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 4013 . . . . . . . . . . 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 15001 . . . . . . . . 9 ((πœ‘ ∧ 𝑅 ∈ V) β†’ 𝑅 βŠ† (t*recβ€˜π‘…))
2714rtrclreclem3 15004 . . . . . . . . . 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 6902 . . . . . . . . . . 11 (t*recβ€˜π‘…) ∈ V
30 sseq2 4008 . . . . . . . . . . . . . 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 4008 . . . . . . . . . . . . . 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 4015 . . . . . . . . . . . . . 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 3662 . . . . . . . . . . 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 4335 . . . . . . 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 5337 . . . . . . 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 7003 . . . . 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 4967 . . . . . . 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 4008 . . . . . . . . . . . 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 4008 . . . . . . . . . . . 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 4015 . . . . . . . . . . . 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 3668 . . . . . . . . . 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 15005 . . . . . . . . . . 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 4968 . . . . . . . 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 3999 . . . . 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 14932 . . . . 5 t* = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
66 fveq1 6888 . . . . . . 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 6881 . . 3 (Β¬ 𝑅 ∈ V β†’ (t*β€˜π‘…) = βˆ…)
73 fvprc 6881 . . . 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 3946   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908  βˆ© cint 4950   ↦ cmpt 5231   I cid 5573  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   ∘ ccom 5680  Rel wrel 5681  β€˜cfv 6541  t*crtcl 14930  t*reccrtrcl 14999
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
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 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-iun 4999  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-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-seq 13964  df-rtrcl 14932  df-relexp 14964  df-rtrclrec 15000
This theorem is referenced by:  rtrclind  15009
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