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Theorem djudom 7091
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 6748 . . 3 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
21adantr 276 . 2 ((𝐴𝐵𝐶𝐷) → ∃𝑓 𝑓:𝐴1-1𝐵)
3 brdomi 6748 . . . 4 (𝐶𝐷 → ∃𝑔 𝑔:𝐶1-1𝐷)
43ad2antlr 489 . . 3 (((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) → ∃𝑔 𝑔:𝐶1-1𝐷)
5 inlresf1 7059 . . . . . . . . 9 (inl ↾ 𝐵):𝐵1-1→(𝐵𝐷)
6 simplr 528 . . . . . . . . 9 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝑓:𝐴1-1𝐵)
7 f1co 5433 . . . . . . . . 9 (((inl ↾ 𝐵):𝐵1-1→(𝐵𝐷) ∧ 𝑓:𝐴1-1𝐵) → ((inl ↾ 𝐵) ∘ 𝑓):𝐴1-1→(𝐵𝐷))
85, 6, 7sylancr 414 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inl ↾ 𝐵) ∘ 𝑓):𝐴1-1→(𝐵𝐷))
9 inrresf1 7060 . . . . . . . . 9 (inr ↾ 𝐷):𝐷1-1→(𝐵𝐷)
10 simpr 110 . . . . . . . . 9 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝑔:𝐶1-1𝐷)
11 f1co 5433 . . . . . . . . 9 (((inr ↾ 𝐷):𝐷1-1→(𝐵𝐷) ∧ 𝑔:𝐶1-1𝐷) → ((inr ↾ 𝐷) ∘ 𝑔):𝐶1-1→(𝐵𝐷))
129, 10, 11sylancr 414 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inr ↾ 𝐷) ∘ 𝑔):𝐶1-1→(𝐵𝐷))
13 rnco 5135 . . . . . . . . . . 11 ran ((inl ↾ 𝐵) ∘ 𝑓) = ran ((inl ↾ 𝐵) ↾ ran 𝑓)
14 f1rn 5422 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
1514ad2antlr 489 . . . . . . . . . . . . 13 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran 𝑓𝐵)
16 resabs1 4936 . . . . . . . . . . . . 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 4856 . . . . . . . . . . 11 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inl ↾ 𝐵) ↾ ran 𝑓) = ran (inl ↾ ran 𝑓))
1913, 18eqtrid 2222 . . . . . . . . . 10 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inl ↾ 𝐵) ∘ 𝑓) = ran (inl ↾ ran 𝑓))
20 rnco 5135 . . . . . . . . . . 11 ran ((inr ↾ 𝐷) ∘ 𝑔) = ran ((inr ↾ 𝐷) ↾ ran 𝑔)
21 f1rn 5422 . . . . . . . . . . . . 13 (𝑔:𝐶1-1𝐷 → ran 𝑔𝐷)
22 resabs1 4936 . . . . . . . . . . . . 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 4856 . . . . . . . . . . 11 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inr ↾ 𝐷) ↾ ran 𝑔) = ran (inr ↾ ran 𝑔))
2520, 24eqtrid 2222 . . . . . . . . . 10 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inr ↾ 𝐷) ∘ 𝑔) = ran (inr ↾ ran 𝑔))
2619, 25ineq12d 3337 . . . . . . . . 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 7061 . . . . . . . . 9 (ran (inl ↾ ran 𝑓) ∩ ran (inr ↾ ran 𝑔)) = ∅
2826, 27eqtrdi 2226 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (ran ((inl ↾ 𝐵) ∘ 𝑓) ∩ ran ((inr ↾ 𝐷) ∘ 𝑔)) = ∅)
298, 12, 28casef1 7088 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷))
30 f1f 5421 . . . . . . 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 6744 . . . . . . . . 9 Rel ≼
3332brrelex1i 4669 . . . . . . . 8 (𝐴𝐵𝐴 ∈ V)
3433ad3antrrr 492 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐴 ∈ V)
3532brrelex1i 4669 . . . . . . . 8 (𝐶𝐷𝐶 ∈ V)
3635ad3antlr 493 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐶 ∈ V)
37 djuex 7041 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐶) ∈ V)
3834, 36, 37syl2anc 411 . . . . . 6 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (𝐴𝐶) ∈ V)
39 fex 5745 . . . . . 6 ((case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)⟶(𝐵𝐷) ∧ (𝐴𝐶) ∈ V) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) ∈ V)
4031, 38, 39syl2anc 411 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) ∈ V)
41 f1eq1 5416 . . . . . 6 ( = case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) → (:(𝐴𝐶)–1-1→(𝐵𝐷) ↔ case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷)))
4241spcegv 2825 . . . . 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 62 . . . 4 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ∃ :(𝐴𝐶)–1-1→(𝐵𝐷))
4432brrelex2i 4670 . . . . . 6 (𝐴𝐵𝐵 ∈ V)
4544ad3antrrr 492 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐵 ∈ V)
4632brrelex2i 4670 . . . . . 6 (𝐶𝐷𝐷 ∈ V)
4746ad3antlr 493 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐷 ∈ V)
48 djuex 7041 . . . . . 6 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
49 brdomg 6747 . . . . . 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 411 . . . 4 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((𝐴𝐶) ≼ (𝐵𝐷) ↔ ∃ :(𝐴𝐶)–1-1→(𝐵𝐷)))
5243, 51mpbird 167 . . 3 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))
534, 52exlimddv 1898 . 2 (((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) → (𝐴𝐶) ≼ (𝐵𝐷))
542, 53exlimddv 1898 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  cin 3128  wss 3129  c0 3422   class class class wbr 4003  ran crn 4627  cres 4628  ccom 4630  wf 5212  1-1wf1 5213  cdom 6738  cdju 7035  inlcinl 7043  inrcinr 7044  casecdjucase 7081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dom 6741  df-dju 7036  df-inl 7045  df-inr 7046  df-case 7082
This theorem is referenced by:  exmidfodomrlemr  7200  exmidfodomrlemrALT  7201  sbthom  14744
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