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Theorem tfsconcatlem 43349
Description: Lemma for tfsconcatun 43350. (Contributed by RP, 23-Feb-2025.)
Assertion
Ref Expression
tfsconcatlem ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥𝑦𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹𝑦)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦

Proof of Theorem tfsconcatlem
StepHypRef Expression
1 onss 7805 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
213ad2ant2 1135 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐵 ⊆ On)
3 oacl 8573 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4 eloni 6394 . . . . . . . . . . . . . . . 16 ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
53, 4syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
6 eloni 6394 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → Ord 𝐴)
76adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
8 ordeldif 43271 . . . . . . . . . . . . . . 15 ((Ord (𝐴 +o 𝐵) ∧ Ord 𝐴) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴𝐶)))
95, 7, 8syl2anc 584 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴𝐶)))
109biimpa 476 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ (𝐴 +o 𝐵) ∧ 𝐴𝐶))
1110ancomd 461 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵)))
1211ex 412 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) → (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))))
1312imdistani 568 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))))
14133impa 1110 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))))
15 oawordex2 43339 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
1614, 15syl 17 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
17 simp1 1137 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐴 ∈ On)
18 onss 7805 . . . . . . . . . . . . . . 15 ((𝐴 +o 𝐵) ∈ On → (𝐴 +o 𝐵) ⊆ On)
193, 18syl 17 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ On)
2019ssdifd 4145 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ⊆ (On ∖ 𝐴))
2120sselda 3983 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
22213impa 1110 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
23 ordon 7797 . . . . . . . . . . . 12 Ord On
2417, 6syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → Ord 𝐴)
25 ordeldif 43271 . . . . . . . . . . . 12 ((Ord On ∧ Ord 𝐴) → (𝐶 ∈ (On ∖ 𝐴) ↔ (𝐶 ∈ On ∧ 𝐴𝐶)))
2623, 24, 25sylancr 587 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ (On ∖ 𝐴) ↔ (𝐶 ∈ On ∧ 𝐴𝐶)))
2722, 26mpbid 232 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (𝐶 ∈ On ∧ 𝐴𝐶))
28 anass 468 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐶) ↔ (𝐴 ∈ On ∧ (𝐶 ∈ On ∧ 𝐴𝐶)))
2917, 27, 28sylanbrc 583 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐶))
30 oawordeu 8593 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐶) → ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶)
3129, 30syl 17 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶)
32 reuss 4327 . . . . . . . 8 ((𝐵 ⊆ On ∧ ∃𝑦𝐵 (𝐴 +o 𝑦) = 𝐶 ∧ ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶) → ∃!𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
332, 16, 31, 32syl3anc 1373 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
34 reurmo 3383 . . . . . . 7 (∃!𝑦𝐵 (𝐴 +o 𝑦) = 𝐶 → ∃*𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
3533, 34syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
36 df-rmo 3380 . . . . . 6 (∃*𝑦𝐵 (𝐴 +o 𝑦) = 𝐶 ↔ ∃*𝑦(𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶))
3735, 36sylib 218 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑦(𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶))
38 moeq 3713 . . . . . 6 ∃*𝑥 𝑥 = (𝐹𝑦)
3938ax-gen 1795 . . . . 5 𝑦∃*𝑥 𝑥 = (𝐹𝑦)
40 moexexvw 2628 . . . . 5 ((∃*𝑦(𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ ∀𝑦∃*𝑥 𝑥 = (𝐹𝑦)) → ∃*𝑥𝑦((𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹𝑦)))
4137, 39, 40sylancl 586 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑥𝑦((𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹𝑦)))
42 df-rex 3071 . . . . . 6 (∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦))))
43 anass 468 . . . . . . 7 (((𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹𝑦)) ↔ (𝑦𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦))))
4443exbii 1848 . . . . . 6 (∃𝑦((𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹𝑦)) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦))))
4542, 44bitr4i 278 . . . . 5 (∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃𝑦((𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹𝑦)))
4645mobii 2548 . . . 4 (∃*𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃*𝑥𝑦((𝑦𝐵 ∧ (𝐴 +o 𝑦) = 𝐶) ∧ 𝑥 = (𝐹𝑦)))
4741, 46sylibr 234 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃*𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)))
48 fvex 6919 . . . . . . . . 9 (𝐹𝑦) ∈ V
4948isseti 3498 . . . . . . . 8 𝑥 𝑥 = (𝐹𝑦)
5049jctr 524 . . . . . . 7 ((𝐴 +o 𝑦) = 𝐶 → ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹𝑦)))
5150a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) ∧ 𝑦𝐵) → ((𝐴 +o 𝑦) = 𝐶 → ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹𝑦))))
5251reximdva 3168 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (∃𝑦𝐵 (𝐴 +o 𝑦) = 𝐶 → ∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹𝑦))))
5316, 52mpd 15 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹𝑦)))
54 rexcom4a 3292 . . . . 5 (∃𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹𝑦)))
55 exmoeu 2581 . . . . 5 (∃𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ (∃*𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) → ∃!𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦))))
5654, 55bitr3i 277 . . . 4 (∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹𝑦)) ↔ (∃*𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) → ∃!𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦))))
5753, 56sylib 218 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → (∃*𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) → ∃!𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦))))
5847, 57mpd 15 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)))
59 eqcom 2744 . . . . 5 ((𝐴 +o 𝑦) = 𝐶𝐶 = (𝐴 +o 𝑦))
6059anbi1i 624 . . . 4 (((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹𝑦)))
6160rexbii 3094 . . 3 (∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃𝑦𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹𝑦)))
6261eubii 2585 . 2 (∃!𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃!𝑥𝑦𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹𝑦)))
6358, 62sylib 218 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥𝑦𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  wrex 3070  ∃!wreu 3378  ∃*wrmo 3379  cdif 3948  wss 3951  Ord word 6383  Oncon0 6384  cfv 6561  (class class class)co 7431   +o coa 8503
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510
This theorem is referenced by:  tfsconcatun  43350  tfsconcatfn  43351  tfsconcatfv1  43352  tfsconcatfv2  43353
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