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Theorem tfsconcatlem 43325
Description: Lemma for tfsconcatun 43326. (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 7803 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ⊆ On)
213ad2ant2 1133 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐵 ⊆ On)
3 oacl 8571 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4 eloni 6395 . . . . . . . . . . . . . . . 16 ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
53, 4syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
6 eloni 6395 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → Ord 𝐴)
76adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
8 ordeldif 43247 . . . . . . . . . . . . . . 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 1109 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))))
15 oawordex2 43315 . . . . . . . . 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 1135 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐴 ∈ On)
18 onss 7803 . . . . . . . . . . . . . . 15 ((𝐴 +o 𝐵) ∈ On → (𝐴 +o 𝐵) ⊆ On)
193, 18syl 17 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ On)
2019ssdifd 4154 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ⊆ (On ∖ 𝐴))
2120sselda 3994 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
22213impa 1109 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → 𝐶 ∈ (On ∖ 𝐴))
23 ordon 7795 . . . . . . . . . . . 12 Ord On
2417, 6syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → Ord 𝐴)
25 ordeldif 43247 . . . . . . . . . . . 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 8591 . . . . . . . . 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 4332 . . . . . . . 8 ((𝐵 ⊆ On ∧ ∃𝑦𝐵 (𝐴 +o 𝑦) = 𝐶 ∧ ∃!𝑦 ∈ On (𝐴 +o 𝑦) = 𝐶) → ∃!𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
332, 16, 31, 32syl3anc 1370 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑦𝐵 (𝐴 +o 𝑦) = 𝐶)
34 reurmo 3380 . . . . . . 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 3377 . . . . . 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 3715 . . . . . 6 ∃*𝑥 𝑥 = (𝐹𝑦)
3938ax-gen 1791 . . . . 5 𝑦∃*𝑥 𝑥 = (𝐹𝑦)
40 moexexvw 2625 . . . . 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 3068 . . . . . 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 1844 . . . . . 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 2545 . . . 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 3495 . . . . . . . 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 3165 . . . . 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 3289 . . . . 5 (∃𝑥𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶 ∧ ∃𝑥 𝑥 = (𝐹𝑦)))
55 exmoeu 2578 . . . . 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 2741 . . . . 5 ((𝐴 +o 𝑦) = 𝐶𝐶 = (𝐴 +o 𝑦))
6059anbi1i 624 . . . 4 (((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹𝑦)))
6160rexbii 3091 . . 3 (∃𝑦𝐵 ((𝐴 +o 𝑦) = 𝐶𝑥 = (𝐹𝑦)) ↔ ∃𝑦𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹𝑦)))
6261eubii 2582 . 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 1086  wal 1534   = wceq 1536  wex 1775  wcel 2105  ∃*wmo 2535  ∃!weu 2565  wrex 3067  ∃!wreu 3375  ∃*wrmo 3376  cdif 3959  wss 3962  Ord word 6384  Oncon0 6385  cfv 6562  (class class class)co 7430   +o coa 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508
This theorem is referenced by:  tfsconcatun  43326  tfsconcatfn  43327  tfsconcatfv1  43328  tfsconcatfv2  43329
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