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Theorem djudom 6787
Description: Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
Assertion
Ref Expression
djudom ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))

Proof of Theorem djudom
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdomi 6466 . . 3 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
21adantr 270 . 2 ((𝐴𝐵𝐶𝐷) → ∃𝑓 𝑓:𝐴1-1𝐵)
3 brdomi 6466 . . . 4 (𝐶𝐷 → ∃𝑔 𝑔:𝐶1-1𝐷)
43ad2antlr 473 . . 3 (((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) → ∃𝑔 𝑔:𝐶1-1𝐷)
5 inlresf1 6753 . . . . . . . . 9 (inl ↾ 𝐵):𝐵1-1→(𝐵𝐷)
6 simplr 497 . . . . . . . . 9 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝑓:𝐴1-1𝐵)
7 f1co 5228 . . . . . . . . 9 (((inl ↾ 𝐵):𝐵1-1→(𝐵𝐷) ∧ 𝑓:𝐴1-1𝐵) → ((inl ↾ 𝐵) ∘ 𝑓):𝐴1-1→(𝐵𝐷))
85, 6, 7sylancr 405 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inl ↾ 𝐵) ∘ 𝑓):𝐴1-1→(𝐵𝐷))
9 inrresf1 6754 . . . . . . . . 9 (inr ↾ 𝐷):𝐷1-1→(𝐵𝐷)
10 simpr 108 . . . . . . . . 9 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝑔:𝐶1-1𝐷)
11 f1co 5228 . . . . . . . . 9 (((inr ↾ 𝐷):𝐷1-1→(𝐵𝐷) ∧ 𝑔:𝐶1-1𝐷) → ((inr ↾ 𝐷) ∘ 𝑔):𝐶1-1→(𝐵𝐷))
129, 10, 11sylancr 405 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inr ↾ 𝐷) ∘ 𝑔):𝐶1-1→(𝐵𝐷))
13 rnco 4937 . . . . . . . . . . 11 ran ((inl ↾ 𝐵) ∘ 𝑓) = ran ((inl ↾ 𝐵) ↾ ran 𝑓)
14 f1rn 5217 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
1514ad2antlr 473 . . . . . . . . . . . . 13 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran 𝑓𝐵)
16 resabs1 4742 . . . . . . . . . . . . 13 (ran 𝑓𝐵 → ((inl ↾ 𝐵) ↾ ran 𝑓) = (inl ↾ ran 𝑓))
1715, 16syl 14 . . . . . . . . . . . 12 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inl ↾ 𝐵) ↾ ran 𝑓) = (inl ↾ ran 𝑓))
1817rneqd 4664 . . . . . . . . . . 11 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inl ↾ 𝐵) ↾ ran 𝑓) = ran (inl ↾ ran 𝑓))
1913, 18syl5eq 2132 . . . . . . . . . 10 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inl ↾ 𝐵) ∘ 𝑓) = ran (inl ↾ ran 𝑓))
20 rnco 4937 . . . . . . . . . . 11 ran ((inr ↾ 𝐷) ∘ 𝑔) = ran ((inr ↾ 𝐷) ↾ ran 𝑔)
21 f1rn 5217 . . . . . . . . . . . . 13 (𝑔:𝐶1-1𝐷 → ran 𝑔𝐷)
22 resabs1 4742 . . . . . . . . . . . . 13 (ran 𝑔𝐷 → ((inr ↾ 𝐷) ↾ ran 𝑔) = (inr ↾ ran 𝑔))
2310, 21, 223syl 17 . . . . . . . . . . . 12 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inr ↾ 𝐷) ↾ ran 𝑔) = (inr ↾ ran 𝑔))
2423rneqd 4664 . . . . . . . . . . 11 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inr ↾ 𝐷) ↾ ran 𝑔) = ran (inr ↾ ran 𝑔))
2520, 24syl5eq 2132 . . . . . . . . . 10 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inr ↾ 𝐷) ∘ 𝑔) = ran (inr ↾ ran 𝑔))
2619, 25ineq12d 3202 . . . . . . . . 9 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (ran ((inl ↾ 𝐵) ∘ 𝑓) ∩ ran ((inr ↾ 𝐷) ∘ 𝑔)) = (ran (inl ↾ ran 𝑓) ∩ ran (inr ↾ ran 𝑔)))
27 djuinr 6755 . . . . . . . . 9 (ran (inl ↾ ran 𝑓) ∩ ran (inr ↾ ran 𝑔)) = ∅
2826, 27syl6eq 2136 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (ran ((inl ↾ 𝐵) ∘ 𝑓) ∩ ran ((inr ↾ 𝐷) ∘ 𝑔)) = ∅)
298, 12, 28casef1 6781 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷))
30 f1f 5216 . . . . . . 7 (case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)⟶(𝐵𝐷))
3129, 30syl 14 . . . . . 6 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)⟶(𝐵𝐷))
32 reldom 6462 . . . . . . . . 9 Rel ≼
3332brrelexi 4479 . . . . . . . 8 (𝐴𝐵𝐴 ∈ V)
3433ad3antrrr 476 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐴 ∈ V)
3532brrelexi 4479 . . . . . . . 8 (𝐶𝐷𝐶 ∈ V)
3635ad3antlr 477 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐶 ∈ V)
37 djuex 6736 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐶) ∈ V)
3834, 36, 37syl2anc 403 . . . . . 6 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (𝐴𝐶) ∈ V)
39 fex 5524 . . . . . 6 ((case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)⟶(𝐵𝐷) ∧ (𝐴𝐶) ∈ V) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) ∈ V)
4031, 38, 39syl2anc 403 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) ∈ V)
41 f1eq1 5211 . . . . . 6 ( = case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) → (:(𝐴𝐶)–1-1→(𝐵𝐷) ↔ case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷)))
4241spcegv 2707 . . . . 5 (case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) ∈ V → (case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷) → ∃ :(𝐴𝐶)–1-1→(𝐵𝐷)))
4340, 29, 42sylc 61 . . . 4 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ∃ :(𝐴𝐶)–1-1→(𝐵𝐷))
4432brrelex2i 4482 . . . . . 6 (𝐴𝐵𝐵 ∈ V)
4544ad3antrrr 476 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐵 ∈ V)
4632brrelex2i 4482 . . . . . 6 (𝐶𝐷𝐷 ∈ V)
4746ad3antlr 477 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐷 ∈ V)
48 djuex 6736 . . . . . 6 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
49 brdomg 6465 . . . . . 6 ((𝐵𝐷) ∈ V → ((𝐴𝐶) ≼ (𝐵𝐷) ↔ ∃ :(𝐴𝐶)–1-1→(𝐵𝐷)))
5048, 49syl 14 . . . . 5 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → ((𝐴𝐶) ≼ (𝐵𝐷) ↔ ∃ :(𝐴𝐶)–1-1→(𝐵𝐷)))
5145, 47, 50syl2anc 403 . . . 4 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((𝐴𝐶) ≼ (𝐵𝐷) ↔ ∃ :(𝐴𝐶)–1-1→(𝐵𝐷)))
5243, 51mpbird 165 . . 3 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))
534, 52exlimddv 1826 . 2 (((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) → (𝐴𝐶) ≼ (𝐵𝐷))
542, 53exlimddv 1826 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  cin 2998  wss 2999  c0 3286   class class class wbr 3845  ran crn 4439  cres 4440  ccom 4442  wf 5011  1-1wf1 5012  cdom 6456  cdju 6730  inlcinl 6737  inrcinr 6738  casecdjucase 6774
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-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
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-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dom 6459  df-dju 6731  df-inl 6739  df-inr 6740  df-case 6775
This theorem is referenced by:  exmidfodomrlemr  6828  exmidfodomrlemrALT  6829
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