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Theorem enomnilem 6773
Description: Lemma for enomni 6774. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
Assertion
Ref Expression
enomnilem (𝐴𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))

Proof of Theorem enomnilem
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 6444 . . . . . . 7 (𝐴𝐵 ↔ ∃ :𝐴1-1-onto𝐵)
21biimpi 118 . . . . . 6 (𝐴𝐵 → ∃ :𝐴1-1-onto𝐵)
32ad2antrr 472 . . . . 5 (((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) → ∃ :𝐴1-1-onto𝐵)
4 fveq1 5288 . . . . . . . . . 10 (𝑓 = (𝑔) → (𝑓𝑥) = ((𝑔)‘𝑥))
54eqeq1d 2096 . . . . . . . . 9 (𝑓 = (𝑔) → ((𝑓𝑥) = ∅ ↔ ((𝑔)‘𝑥) = ∅))
65rexbidv 2381 . . . . . . . 8 (𝑓 = (𝑔) → (∃𝑥𝐴 (𝑓𝑥) = ∅ ↔ ∃𝑥𝐴 ((𝑔)‘𝑥) = ∅))
74eqeq1d 2096 . . . . . . . . 9 (𝑓 = (𝑔) → ((𝑓𝑥) = 1𝑜 ↔ ((𝑔)‘𝑥) = 1𝑜))
87ralbidv 2380 . . . . . . . 8 (𝑓 = (𝑔) → (∀𝑥𝐴 (𝑓𝑥) = 1𝑜 ↔ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜))
96, 8orbi12d 742 . . . . . . 7 (𝑓 = (𝑔) → ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜) ↔ (∃𝑥𝐴 ((𝑔)‘𝑥) = ∅ ∨ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜)))
10 isomnimap 6772 . . . . . . . . 9 (𝐴 ∈ Omni → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ (2𝑜𝑚 𝐴)(∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜)))
1110ibi 174 . . . . . . . 8 (𝐴 ∈ Omni → ∀𝑓 ∈ (2𝑜𝑚 𝐴)(∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜))
1211ad3antlr 477 . . . . . . 7 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → ∀𝑓 ∈ (2𝑜𝑚 𝐴)(∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜))
13 simpr 108 . . . . . . . . . . 11 (((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) → 𝑔 ∈ (2𝑜𝑚 𝐵))
14 2onn 6260 . . . . . . . . . . . . 13 2𝑜 ∈ ω
15 relen 6441 . . . . . . . . . . . . . 14 Rel ≈
1615brrelex2i 4471 . . . . . . . . . . . . 13 (𝐴𝐵𝐵 ∈ V)
17 elmapg 6398 . . . . . . . . . . . . 13 ((2𝑜 ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2𝑜𝑚 𝐵) ↔ 𝑔:𝐵⟶2𝑜))
1814, 16, 17sylancr 405 . . . . . . . . . . . 12 (𝐴𝐵 → (𝑔 ∈ (2𝑜𝑚 𝐵) ↔ 𝑔:𝐵⟶2𝑜))
1918ad2antrr 472 . . . . . . . . . . 11 (((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) → (𝑔 ∈ (2𝑜𝑚 𝐵) ↔ 𝑔:𝐵⟶2𝑜))
2013, 19mpbid 145 . . . . . . . . . 10 (((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) → 𝑔:𝐵⟶2𝑜)
2120adantr 270 . . . . . . . . 9 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → 𝑔:𝐵⟶2𝑜)
22 f1of 5237 . . . . . . . . . 10 (:𝐴1-1-onto𝐵:𝐴𝐵)
2322adantl 271 . . . . . . . . 9 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → :𝐴𝐵)
24 fco 5161 . . . . . . . . 9 ((𝑔:𝐵⟶2𝑜:𝐴𝐵) → (𝑔):𝐴⟶2𝑜)
2521, 23, 24syl2anc 403 . . . . . . . 8 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → (𝑔):𝐴⟶2𝑜)
26 simpllr 501 . . . . . . . . 9 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → 𝐴 ∈ Omni)
27 elmapg 6398 . . . . . . . . 9 ((2𝑜 ∈ ω ∧ 𝐴 ∈ Omni) → ((𝑔) ∈ (2𝑜𝑚 𝐴) ↔ (𝑔):𝐴⟶2𝑜))
2814, 26, 27sylancr 405 . . . . . . . 8 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → ((𝑔) ∈ (2𝑜𝑚 𝐴) ↔ (𝑔):𝐴⟶2𝑜))
2925, 28mpbird 165 . . . . . . 7 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → (𝑔) ∈ (2𝑜𝑚 𝐴))
309, 12, 29rspcdva 2727 . . . . . 6 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → (∃𝑥𝐴 ((𝑔)‘𝑥) = ∅ ∨ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜))
31 f1ofn 5238 . . . . . . . . . . . 12 (:𝐴1-1-onto𝐵 Fn 𝐴)
3231ad2antlr 473 . . . . . . . . . . 11 (((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) → Fn 𝐴)
33 fvco2 5357 . . . . . . . . . . 11 (( Fn 𝐴𝑥𝐴) → ((𝑔)‘𝑥) = (𝑔‘(𝑥)))
3432, 33sylancom 411 . . . . . . . . . 10 (((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) → ((𝑔)‘𝑥) = (𝑔‘(𝑥)))
3534eqeq1d 2096 . . . . . . . . 9 (((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) → (((𝑔)‘𝑥) = ∅ ↔ (𝑔‘(𝑥)) = ∅))
3623ffvelrnda 5418 . . . . . . . . . 10 (((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) → (𝑥) ∈ 𝐵)
37 simpr 108 . . . . . . . . . . . 12 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) ∧ 𝑦 = (𝑥)) → 𝑦 = (𝑥))
3837fveq2d 5293 . . . . . . . . . . 11 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) ∧ 𝑦 = (𝑥)) → (𝑔𝑦) = (𝑔‘(𝑥)))
3938eqeq1d 2096 . . . . . . . . . 10 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) ∧ 𝑦 = (𝑥)) → ((𝑔𝑦) = ∅ ↔ (𝑔‘(𝑥)) = ∅))
4036, 39rspcedv 2726 . . . . . . . . 9 (((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) → ((𝑔‘(𝑥)) = ∅ → ∃𝑦𝐵 (𝑔𝑦) = ∅))
4135, 40sylbid 148 . . . . . . . 8 (((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ 𝑥𝐴) → (((𝑔)‘𝑥) = ∅ → ∃𝑦𝐵 (𝑔𝑦) = ∅))
4241rexlimdva 2489 . . . . . . 7 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → (∃𝑥𝐴 ((𝑔)‘𝑥) = ∅ → ∃𝑦𝐵 (𝑔𝑦) = ∅))
4331ad3antlr 477 . . . . . . . . . . 11 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → Fn 𝐴)
44 f1ocnv 5250 . . . . . . . . . . . . . 14 (:𝐴1-1-onto𝐵:𝐵1-1-onto𝐴)
45 f1of 5237 . . . . . . . . . . . . . 14 (:𝐵1-1-onto𝐴:𝐵𝐴)
4644, 45syl 14 . . . . . . . . . . . . 13 (:𝐴1-1-onto𝐵:𝐵𝐴)
4746ad3antlr 477 . . . . . . . . . . . 12 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → :𝐵𝐴)
48 simpr 108 . . . . . . . . . . . 12 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → 𝑦𝐵)
4947, 48ffvelrnd 5419 . . . . . . . . . . 11 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → (𝑦) ∈ 𝐴)
50 fvco2 5357 . . . . . . . . . . 11 (( Fn 𝐴 ∧ (𝑦) ∈ 𝐴) → ((𝑔)‘(𝑦)) = (𝑔‘(‘(𝑦))))
5143, 49, 50syl2anc 403 . . . . . . . . . 10 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → ((𝑔)‘(𝑦)) = (𝑔‘(‘(𝑦))))
52 fveq2 5289 . . . . . . . . . . . 12 (𝑥 = (𝑦) → ((𝑔)‘𝑥) = ((𝑔)‘(𝑦)))
5352eqeq1d 2096 . . . . . . . . . . 11 (𝑥 = (𝑦) → (((𝑔)‘𝑥) = 1𝑜 ↔ ((𝑔)‘(𝑦)) = 1𝑜))
54 simplr 497 . . . . . . . . . . 11 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜)
5553, 54, 49rspcdva 2727 . . . . . . . . . 10 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → ((𝑔)‘(𝑦)) = 1𝑜)
56 simpllr 501 . . . . . . . . . . 11 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → :𝐴1-1-onto𝐵)
57 f1ocnvfv2 5539 . . . . . . . . . . . 12 ((:𝐴1-1-onto𝐵𝑦𝐵) → (‘(𝑦)) = 𝑦)
5857fveq2d 5293 . . . . . . . . . . 11 ((:𝐴1-1-onto𝐵𝑦𝐵) → (𝑔‘(‘(𝑦))) = (𝑔𝑦))
5956, 58sylancom 411 . . . . . . . . . 10 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → (𝑔‘(‘(𝑦))) = (𝑔𝑦))
6051, 55, 593eqtr3rd 2129 . . . . . . . . 9 ((((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) ∧ 𝑦𝐵) → (𝑔𝑦) = 1𝑜)
6160ralrimiva 2446 . . . . . . . 8 (((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) ∧ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) → ∀𝑦𝐵 (𝑔𝑦) = 1𝑜)
6261ex 113 . . . . . . 7 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → (∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜 → ∀𝑦𝐵 (𝑔𝑦) = 1𝑜))
6342, 62orim12d 735 . . . . . 6 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → ((∃𝑥𝐴 ((𝑔)‘𝑥) = ∅ ∨ ∀𝑥𝐴 ((𝑔)‘𝑥) = 1𝑜) → (∃𝑦𝐵 (𝑔𝑦) = ∅ ∨ ∀𝑦𝐵 (𝑔𝑦) = 1𝑜)))
6430, 63mpd 13 . . . . 5 ((((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) ∧ :𝐴1-1-onto𝐵) → (∃𝑦𝐵 (𝑔𝑦) = ∅ ∨ ∀𝑦𝐵 (𝑔𝑦) = 1𝑜))
653, 64exlimddv 1826 . . . 4 (((𝐴𝐵𝐴 ∈ Omni) ∧ 𝑔 ∈ (2𝑜𝑚 𝐵)) → (∃𝑦𝐵 (𝑔𝑦) = ∅ ∨ ∀𝑦𝐵 (𝑔𝑦) = 1𝑜))
6665ralrimiva 2446 . . 3 ((𝐴𝐵𝐴 ∈ Omni) → ∀𝑔 ∈ (2𝑜𝑚 𝐵)(∃𝑦𝐵 (𝑔𝑦) = ∅ ∨ ∀𝑦𝐵 (𝑔𝑦) = 1𝑜))
67 isomnimap 6772 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ Omni ↔ ∀𝑔 ∈ (2𝑜𝑚 𝐵)(∃𝑦𝐵 (𝑔𝑦) = ∅ ∨ ∀𝑦𝐵 (𝑔𝑦) = 1𝑜)))
6816, 67syl 14 . . . 4 (𝐴𝐵 → (𝐵 ∈ Omni ↔ ∀𝑔 ∈ (2𝑜𝑚 𝐵)(∃𝑦𝐵 (𝑔𝑦) = ∅ ∨ ∀𝑦𝐵 (𝑔𝑦) = 1𝑜)))
6968adantr 270 . . 3 ((𝐴𝐵𝐴 ∈ Omni) → (𝐵 ∈ Omni ↔ ∀𝑔 ∈ (2𝑜𝑚 𝐵)(∃𝑦𝐵 (𝑔𝑦) = ∅ ∨ ∀𝑦𝐵 (𝑔𝑦) = 1𝑜)))
7066, 69mpbird 165 . 2 ((𝐴𝐵𝐴 ∈ Omni) → 𝐵 ∈ Omni)
7170ex 113 1 (𝐴𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 664   = wceq 1289  wex 1426  wcel 1438  wral 2359  wrex 2360  Vcvv 2619  c0 3284   class class class wbr 3837  ωcom 4395  ccnv 4427  ccom 4432   Fn wfn 4997  wf 4998  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  1𝑜c1o 6156  2𝑜c2o 6157  𝑚 cmap 6385  cen 6435  Omnicomni 6767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-en 6438  df-omni 6769
This theorem is referenced by:  enomni  6774
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