Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tfsconcatun Structured version   Visualization version   GIF version

Theorem tfsconcatun 43299
Description: The concatenation of two transfinite series is a union of functions. (Contributed by RP, 23-Feb-2025.)
Hypothesis
Ref Expression
tfsconcat.op + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
Assertion
Ref Expression
tfsconcatun (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧   𝐶,𝑎,𝑏,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑥,𝑦,𝑧
Allowed substitution hints:   + (𝑥,𝑦,𝑧,𝑎,𝑏)

Proof of Theorem tfsconcatun
StepHypRef Expression
1 tfsconcat.op . . 3 + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
21a1i 11 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))})))
3 simprl 770 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑎 = 𝐴)
4 dmeq 5928 . . . . . . . . . 10 (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
54adantr 480 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐵) → dom 𝑎 = dom 𝐴)
6 fndm 6682 . . . . . . . . . . 11 (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
76adantr 480 . . . . . . . . . 10 ((𝐴 Fn 𝐶𝐵 Fn 𝐷) → dom 𝐴 = 𝐶)
87adantr 480 . . . . . . . . 9 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐴 = 𝐶)
95, 8sylan9eqr 2802 . . . . . . . 8 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → dom 𝑎 = 𝐶)
10 dmeq 5928 . . . . . . . . . 10 (𝑏 = 𝐵 → dom 𝑏 = dom 𝐵)
1110adantl 481 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐵) → dom 𝑏 = dom 𝐵)
12 fndm 6682 . . . . . . . . . . 11 (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
1312adantl 481 . . . . . . . . . 10 ((𝐴 Fn 𝐶𝐵 Fn 𝐷) → dom 𝐵 = 𝐷)
1413adantr 480 . . . . . . . . 9 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐵 = 𝐷)
1511, 14sylan9eqr 2802 . . . . . . . 8 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → dom 𝑏 = 𝐷)
169, 15oveq12d 7466 . . . . . . 7 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (dom 𝑎 +o dom 𝑏) = (𝐶 +o 𝐷))
1716, 9difeq12d 4150 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) = ((𝐶 +o 𝐷) ∖ 𝐶))
1817eleq2d 2830 . . . . 5 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ↔ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
199oveq1d 7463 . . . . . . . 8 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (dom 𝑎 +o 𝑧) = (𝐶 +o 𝑧))
2019eqeq2d 2751 . . . . . . 7 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑥 = (dom 𝑎 +o 𝑧) ↔ 𝑥 = (𝐶 +o 𝑧)))
21 fveq1 6919 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏𝑧) = (𝐵𝑧))
2221eqeq2d 2751 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑦 = (𝑏𝑧) ↔ 𝑦 = (𝐵𝑧)))
2322adantl 481 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑦 = (𝑏𝑧) ↔ 𝑦 = (𝐵𝑧)))
2423adantl 481 . . . . . . 7 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑦 = (𝑏𝑧) ↔ 𝑦 = (𝐵𝑧)))
2520, 24anbi12d 631 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)) ↔ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))))
2615, 25rexeqbidv 3355 . . . . 5 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)) ↔ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))))
2718, 26anbi12d 631 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧))) ↔ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))))
2827opabbidv 5232 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))})
293, 28uneq12d 4192 . 2 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}))
30 fnex 7254 . . 3 ((𝐴 Fn 𝐶𝐶 ∈ On) → 𝐴 ∈ V)
3130ad2ant2r 746 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐴 ∈ V)
32 fnex 7254 . . 3 ((𝐵 Fn 𝐷𝐷 ∈ On) → 𝐵 ∈ V)
3332ad2ant2l 745 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐵 ∈ V)
34 oacl 8591 . . . . . 6 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
3534difexd 5349 . . . . 5 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ((𝐶 +o 𝐷) ∖ 𝐶) ∈ V)
3635adantl 481 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐶 +o 𝐷) ∖ 𝐶) ∈ V)
37 simplrl 776 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐶 ∈ On)
38 simplrr 777 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐷 ∈ On)
39 simpr 484 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
40 tfsconcatlem 43298 . . . . . 6 ((𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
4137, 38, 39, 40syl3anc 1371 . . . . 5 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
42 euabex 5481 . . . . 5 (∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)) → {𝑦 ∣ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))} ∈ V)
4341, 42syl 17 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → {𝑦 ∣ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))} ∈ V)
4436, 43opabex3d 8006 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} ∈ V)
4531, 44unexd 7789 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}) ∈ V)
462, 29, 31, 33, 45ovmpod 7602 1 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  ∃!weu 2571  {cab 2717  wrex 3076  Vcvv 3488  cdif 3973  cun 3974  {copab 5228  dom cdm 5700  Oncon0 6395   Fn wfn 6568  cfv 6573  (class class class)co 7448  cmpo 7450   +o coa 8519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526
This theorem is referenced by:  tfsconcatfn  43300  tfsconcatfv1  43301  tfsconcatfv2  43302  tfsconcatrn  43304  tfsconcatb0  43306  tfsconcat0i  43307  tfsconcatrev  43310
  Copyright terms: Public domain W3C validator