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Theorem clcnvlem 42676
Description: When 𝐴, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
Hypotheses
Ref Expression
clcnvlem.sub1 ((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → (𝜒𝜓))
clcnvlem.sub2 ((𝜑𝑦 = 𝑥) → (𝜓𝜒))
clcnvlem.sub3 (𝑥 = 𝐴 → (𝜓𝜃))
clcnvlem.ssub (𝜑𝑋𝐴)
clcnvlem.ubex (𝜑𝐴 ∈ V)
clcnvlem.clex (𝜑𝜃)
Assertion
Ref Expression
clcnvlem (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} = {𝑦 ∣ (𝑋𝑦𝜒)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑋   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)

Proof of Theorem clcnvlem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 clcnvlem.ubex . . . 4 (𝜑𝐴 ∈ V)
2 clcnvlem.ssub . . . . 5 (𝜑𝑋𝐴)
3 clcnvlem.clex . . . . 5 (𝜑𝜃)
42, 3jca 510 . . . 4 (𝜑 → (𝑋𝐴𝜃))
5 clcnvlem.sub3 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜃))
65cleq2lem 42661 . . . 4 (𝑥 = 𝐴 → ((𝑋𝑥𝜓) ↔ (𝑋𝐴𝜃)))
71, 4, 6spcedv 3587 . . 3 (𝜑 → ∃𝑥(𝑋𝑥𝜓))
87cnvintabd 42656 . 2 (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} = {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))})
9 df-rab 3431 . . . . 5 {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))} = {𝑧 ∣ (𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)))}
10 exsimpl 1869 . . . . . . . . . . 11 (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) → ∃𝑥 𝑧 = 𝑥)
11 relcnv 6102 . . . . . . . . . . . . 13 Rel 𝑥
12 releq 5775 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (Rel 𝑧 ↔ Rel 𝑥))
1311, 12mpbiri 257 . . . . . . . . . . . 12 (𝑧 = 𝑥 → Rel 𝑧)
1413exlimiv 1931 . . . . . . . . . . 11 (∃𝑥 𝑧 = 𝑥 → Rel 𝑧)
1510, 14syl 17 . . . . . . . . . 10 (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) → Rel 𝑧)
16 df-rel 5682 . . . . . . . . . 10 (Rel 𝑧𝑧 ⊆ (V × V))
1715, 16sylib 217 . . . . . . . . 9 (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) → 𝑧 ⊆ (V × V))
18 velpw 4606 . . . . . . . . . 10 (𝑧 ∈ 𝒫 (V × V) ↔ 𝑧 ⊆ (V × V))
1918bicomi 223 . . . . . . . . 9 (𝑧 ⊆ (V × V) ↔ 𝑧 ∈ 𝒫 (V × V))
2017, 19sylib 217 . . . . . . . 8 (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) → 𝑧 ∈ 𝒫 (V × V))
2120pm4.71ri 559 . . . . . . 7 (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) ↔ (𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))))
2221bicomi 223 . . . . . 6 ((𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))) ↔ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)))
2322abbii 2800 . . . . 5 {𝑧 ∣ (𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)))} = {𝑧 ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))}
249, 23eqtri 2758 . . . 4 {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))} = {𝑧 ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))}
2524inteqi 4953 . . 3 {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))} = {𝑧 ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))}
2625a1i 11 . 2 (𝜑 {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))} = {𝑧 ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))})
27 vex 3476 . . . . . . 7 𝑦 ∈ V
2827cnvex 7918 . . . . . 6 𝑦 ∈ V
2928cnvex 7918 . . . . 5 𝑦 ∈ V
3029a1i 11 . . . 4 (𝜑𝑦 ∈ V)
311, 2ssexd 5323 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
3231difexd 5328 . . . . . . . . . 10 (𝜑 → (𝑋𝑋) ∈ V)
33 unexg 7738 . . . . . . . . . 10 ((𝑦 ∈ V ∧ (𝑋𝑋) ∈ V) → (𝑦 ∪ (𝑋𝑋)) ∈ V)
3428, 32, 33sylancr 585 . . . . . . . . 9 (𝜑 → (𝑦 ∪ (𝑋𝑋)) ∈ V)
35 inundif 4477 . . . . . . . . . . . . . 14 ((𝑋𝑋) ∪ (𝑋𝑋)) = 𝑋
36 cnvun 6141 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋𝑋) ∪ (𝑋𝑋)) = ((𝑋𝑋) ∪ (𝑋𝑋))
3736sseq1i 4009 . . . . . . . . . . . . . . . . . . . 20 (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦 ↔ ((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦)
3837biimpi 215 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦 → ((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦)
3938unssad 4186 . . . . . . . . . . . . . . . . . 18 (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦(𝑋𝑋) ⊆ 𝑦)
40 relcnv 6102 . . . . . . . . . . . . . . . . . . . . 21 Rel 𝑋
41 relin2 5812 . . . . . . . . . . . . . . . . . . . . 21 (Rel 𝑋 → Rel (𝑋𝑋))
4240, 41ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 Rel (𝑋𝑋)
43 dfrel2 6187 . . . . . . . . . . . . . . . . . . . 20 (Rel (𝑋𝑋) ↔ (𝑋𝑋) = (𝑋𝑋))
4442, 43mpbi 229 . . . . . . . . . . . . . . . . . . 19 (𝑋𝑋) = (𝑋𝑋)
45 cnvss 5871 . . . . . . . . . . . . . . . . . . 19 ((𝑋𝑋) ⊆ 𝑦(𝑋𝑋) ⊆ 𝑦)
4644, 45eqsstrrid 4030 . . . . . . . . . . . . . . . . . 18 ((𝑋𝑋) ⊆ 𝑦 → (𝑋𝑋) ⊆ 𝑦)
4739, 46syl 17 . . . . . . . . . . . . . . . . 17 (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦 → (𝑋𝑋) ⊆ 𝑦)
48 ssid 4003 . . . . . . . . . . . . . . . . 17 (𝑋𝑋) ⊆ (𝑋𝑋)
49 unss12 4181 . . . . . . . . . . . . . . . . 17 (((𝑋𝑋) ⊆ 𝑦 ∧ (𝑋𝑋) ⊆ (𝑋𝑋)) → ((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ (𝑦 ∪ (𝑋𝑋)))
5047, 48, 49sylancl 584 . . . . . . . . . . . . . . . 16 (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦 → ((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ (𝑦 ∪ (𝑋𝑋)))
5150a1i 11 . . . . . . . . . . . . . . 15 (((𝑋𝑋) ∪ (𝑋𝑋)) = 𝑋 → (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦 → ((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ (𝑦 ∪ (𝑋𝑋))))
52 cnveq 5872 . . . . . . . . . . . . . . . 16 (((𝑋𝑋) ∪ (𝑋𝑋)) = 𝑋((𝑋𝑋) ∪ (𝑋𝑋)) = 𝑋)
5352sseq1d 4012 . . . . . . . . . . . . . . 15 (((𝑋𝑋) ∪ (𝑋𝑋)) = 𝑋 → (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ 𝑦𝑋𝑦))
54 sseq1 4006 . . . . . . . . . . . . . . 15 (((𝑋𝑋) ∪ (𝑋𝑋)) = 𝑋 → (((𝑋𝑋) ∪ (𝑋𝑋)) ⊆ (𝑦 ∪ (𝑋𝑋)) ↔ 𝑋 ⊆ (𝑦 ∪ (𝑋𝑋))))
5551, 53, 543imtr3d 292 . . . . . . . . . . . . . 14 (((𝑋𝑋) ∪ (𝑋𝑋)) = 𝑋 → (𝑋𝑦𝑋 ⊆ (𝑦 ∪ (𝑋𝑋))))
5635, 55ax-mp 5 . . . . . . . . . . . . 13 (𝑋𝑦𝑋 ⊆ (𝑦 ∪ (𝑋𝑋)))
57 sseq2 4007 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (𝑋𝑥𝑋 ⊆ (𝑦 ∪ (𝑋𝑋))))
5856, 57imbitrrid 245 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (𝑋𝑦𝑋𝑥))
5958adantl 480 . . . . . . . . . . 11 ((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → (𝑋𝑦𝑋𝑥))
60 clcnvlem.sub1 . . . . . . . . . . 11 ((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → (𝜒𝜓))
6159, 60anim12d 607 . . . . . . . . . 10 ((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → ((𝑋𝑦𝜒) → (𝑋𝑥𝜓)))
62 cnvun 6141 . . . . . . . . . . . . 13 (𝑦 ∪ (𝑋𝑋)) = (𝑦(𝑋𝑋))
63 cnvnonrel 42641 . . . . . . . . . . . . . . 15 (𝑋𝑋) = ∅
64 0ss 4395 . . . . . . . . . . . . . . 15 ∅ ⊆ 𝑦
6563, 64eqsstri 4015 . . . . . . . . . . . . . 14 (𝑋𝑋) ⊆ 𝑦
66 ssequn2 4182 . . . . . . . . . . . . . 14 ((𝑋𝑋) ⊆ 𝑦 ↔ (𝑦(𝑋𝑋)) = 𝑦)
6765, 66mpbi 229 . . . . . . . . . . . . 13 (𝑦(𝑋𝑋)) = 𝑦
6862, 67eqtr2i 2759 . . . . . . . . . . . 12 𝑦 = (𝑦 ∪ (𝑋𝑋))
69 cnveq 5872 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → 𝑥 = (𝑦 ∪ (𝑋𝑋)))
7068, 69eqtr4id 2789 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → 𝑦 = 𝑥)
7170adantl 480 . . . . . . . . . 10 ((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → 𝑦 = 𝑥)
7261, 71jctild 524 . . . . . . . . 9 ((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → ((𝑋𝑦𝜒) → (𝑦 = 𝑥 ∧ (𝑋𝑥𝜓))))
7334, 72spcimedv 3584 . . . . . . . 8 (𝜑 → ((𝑋𝑦𝜒) → ∃𝑥(𝑦 = 𝑥 ∧ (𝑋𝑥𝜓))))
7473imp 405 . . . . . . 7 ((𝜑 ∧ (𝑋𝑦𝜒)) → ∃𝑥(𝑦 = 𝑥 ∧ (𝑋𝑥𝜓)))
7574adantlr 711 . . . . . 6 (((𝜑𝑧 = 𝑦) ∧ (𝑋𝑦𝜒)) → ∃𝑥(𝑦 = 𝑥 ∧ (𝑋𝑥𝜓)))
76 eqeq1 2734 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 = 𝑥𝑦 = 𝑥))
7776anbi1d 628 . . . . . . . 8 (𝑧 = 𝑦 → ((𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) ↔ (𝑦 = 𝑥 ∧ (𝑋𝑥𝜓))))
7877exbidv 1922 . . . . . . 7 (𝑧 = 𝑦 → (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) ↔ ∃𝑥(𝑦 = 𝑥 ∧ (𝑋𝑥𝜓))))
7978ad2antlr 723 . . . . . 6 (((𝜑𝑧 = 𝑦) ∧ (𝑋𝑦𝜒)) → (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) ↔ ∃𝑥(𝑦 = 𝑥 ∧ (𝑋𝑥𝜓))))
8075, 79mpbird 256 . . . . 5 (((𝜑𝑧 = 𝑦) ∧ (𝑋𝑦𝜒)) → ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)))
8180ex 411 . . . 4 ((𝜑𝑧 = 𝑦) → ((𝑋𝑦𝜒) → ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))))
82 cnvcnvss 6192 . . . . 5 𝑦𝑦
8382a1i 11 . . . 4 (𝜑𝑦𝑦)
8430, 81, 83intabssd 42572 . . 3 (𝜑 {𝑧 ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))} ⊆ {𝑦 ∣ (𝑋𝑦𝜒)})
85 vex 3476 . . . . 5 𝑧 ∈ V
8685a1i 11 . . . 4 (𝜑𝑧 ∈ V)
87 eqtr 2753 . . . . . . . 8 ((𝑦 = 𝑧𝑧 = 𝑥) → 𝑦 = 𝑥)
88 cnvss 5871 . . . . . . . . . . . 12 (𝑋𝑥𝑋𝑥)
89 sseq2 4007 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑋𝑦𝑋𝑥))
9088, 89imbitrrid 245 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑋𝑥𝑋𝑦))
9190adantl 480 . . . . . . . . . 10 ((𝜑𝑦 = 𝑥) → (𝑋𝑥𝑋𝑦))
92 clcnvlem.sub2 . . . . . . . . . 10 ((𝜑𝑦 = 𝑥) → (𝜓𝜒))
9391, 92anim12d 607 . . . . . . . . 9 ((𝜑𝑦 = 𝑥) → ((𝑋𝑥𝜓) → (𝑋𝑦𝜒)))
9493ex 411 . . . . . . . 8 (𝜑 → (𝑦 = 𝑥 → ((𝑋𝑥𝜓) → (𝑋𝑦𝜒))))
9587, 94syl5 34 . . . . . . 7 (𝜑 → ((𝑦 = 𝑧𝑧 = 𝑥) → ((𝑋𝑥𝜓) → (𝑋𝑦𝜒))))
9695impl 454 . . . . . 6 (((𝜑𝑦 = 𝑧) ∧ 𝑧 = 𝑥) → ((𝑋𝑥𝜓) → (𝑋𝑦𝜒)))
9796expimpd 452 . . . . 5 ((𝜑𝑦 = 𝑧) → ((𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) → (𝑋𝑦𝜒)))
9897exlimdv 1934 . . . 4 ((𝜑𝑦 = 𝑧) → (∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓)) → (𝑋𝑦𝜒)))
99 ssid 4003 . . . . 5 𝑧𝑧
10099a1i 11 . . . 4 (𝜑𝑧𝑧)
10186, 98, 100intabssd 42572 . . 3 (𝜑 {𝑦 ∣ (𝑋𝑦𝜒)} ⊆ {𝑧 ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))})
10284, 101eqssd 3998 . 2 (𝜑 {𝑧 ∣ ∃𝑥(𝑧 = 𝑥 ∧ (𝑋𝑥𝜓))} = {𝑦 ∣ (𝑋𝑦𝜒)})
1038, 26, 1023eqtrd 2774 1 (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} = {𝑦 ∣ (𝑋𝑦𝜒)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  {cab 2707  {crab 3430  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  𝒫 cpw 4601   cint 4949   × cxp 5673  ccnv 5674  Rel wrel 5680
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-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  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-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  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-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-1st 7977  df-2nd 7978
This theorem is referenced by:  cnvtrucl0  42677  cnvrcl0  42678  cnvtrcl0  42679
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