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Theorem djudom 6944
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 6609 . . 3 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
21adantr 272 . 2 ((𝐴𝐵𝐶𝐷) → ∃𝑓 𝑓:𝐴1-1𝐵)
3 brdomi 6609 . . . 4 (𝐶𝐷 → ∃𝑔 𝑔:𝐶1-1𝐷)
43ad2antlr 478 . . 3 (((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) → ∃𝑔 𝑔:𝐶1-1𝐷)
5 inlresf1 6912 . . . . . . . . 9 (inl ↾ 𝐵):𝐵1-1→(𝐵𝐷)
6 simplr 502 . . . . . . . . 9 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝑓:𝐴1-1𝐵)
7 f1co 5308 . . . . . . . . 9 (((inl ↾ 𝐵):𝐵1-1→(𝐵𝐷) ∧ 𝑓:𝐴1-1𝐵) → ((inl ↾ 𝐵) ∘ 𝑓):𝐴1-1→(𝐵𝐷))
85, 6, 7sylancr 408 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inl ↾ 𝐵) ∘ 𝑓):𝐴1-1→(𝐵𝐷))
9 inrresf1 6913 . . . . . . . . 9 (inr ↾ 𝐷):𝐷1-1→(𝐵𝐷)
10 simpr 109 . . . . . . . . 9 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝑔:𝐶1-1𝐷)
11 f1co 5308 . . . . . . . . 9 (((inr ↾ 𝐷):𝐷1-1→(𝐵𝐷) ∧ 𝑔:𝐶1-1𝐷) → ((inr ↾ 𝐷) ∘ 𝑔):𝐶1-1→(𝐵𝐷))
129, 10, 11sylancr 408 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((inr ↾ 𝐷) ∘ 𝑔):𝐶1-1→(𝐵𝐷))
13 rnco 5013 . . . . . . . . . . 11 ran ((inl ↾ 𝐵) ∘ 𝑓) = ran ((inl ↾ 𝐵) ↾ ran 𝑓)
14 f1rn 5297 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
1514ad2antlr 478 . . . . . . . . . . . . 13 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran 𝑓𝐵)
16 resabs1 4816 . . . . . . . . . . . . 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 4736 . . . . . . . . . . 11 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inl ↾ 𝐵) ↾ ran 𝑓) = ran (inl ↾ ran 𝑓))
1913, 18syl5eq 2160 . . . . . . . . . 10 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inl ↾ 𝐵) ∘ 𝑓) = ran (inl ↾ ran 𝑓))
20 rnco 5013 . . . . . . . . . . 11 ran ((inr ↾ 𝐷) ∘ 𝑔) = ran ((inr ↾ 𝐷) ↾ ran 𝑔)
21 f1rn 5297 . . . . . . . . . . . . 13 (𝑔:𝐶1-1𝐷 → ran 𝑔𝐷)
22 resabs1 4816 . . . . . . . . . . . . 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 4736 . . . . . . . . . . 11 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inr ↾ 𝐷) ↾ ran 𝑔) = ran (inr ↾ ran 𝑔))
2520, 24syl5eq 2160 . . . . . . . . . 10 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ran ((inr ↾ 𝐷) ∘ 𝑔) = ran (inr ↾ ran 𝑔))
2619, 25ineq12d 3246 . . . . . . . . 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 6914 . . . . . . . . 9 (ran (inl ↾ ran 𝑓) ∩ ran (inr ↾ ran 𝑔)) = ∅
2826, 27syl6eq 2164 . . . . . . . 8 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (ran ((inl ↾ 𝐵) ∘ 𝑓) ∩ ran ((inr ↾ 𝐷) ∘ 𝑔)) = ∅)
298, 12, 28casef1 6941 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷))
30 f1f 5296 . . . . . . 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 6605 . . . . . . . . 9 Rel ≼
3332brrelex1i 4550 . . . . . . . 8 (𝐴𝐵𝐴 ∈ V)
3433ad3antrrr 481 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐴 ∈ V)
3532brrelex1i 4550 . . . . . . . 8 (𝐶𝐷𝐶 ∈ V)
3635ad3antlr 482 . . . . . . 7 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐶 ∈ V)
37 djuex 6894 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐶) ∈ V)
3834, 36, 37syl2anc 406 . . . . . 6 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (𝐴𝐶) ∈ V)
39 fex 5613 . . . . . 6 ((case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)⟶(𝐵𝐷) ∧ (𝐴𝐶) ∈ V) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) ∈ V)
4031, 38, 39syl2anc 406 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) ∈ V)
41 f1eq1 5291 . . . . . 6 ( = case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)) → (:(𝐴𝐶)–1-1→(𝐵𝐷) ↔ case(((inl ↾ 𝐵) ∘ 𝑓), ((inr ↾ 𝐷) ∘ 𝑔)):(𝐴𝐶)–1-1→(𝐵𝐷)))
4241spcegv 2746 . . . . 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 4551 . . . . . 6 (𝐴𝐵𝐵 ∈ V)
4544ad3antrrr 481 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐵 ∈ V)
4632brrelex2i 4551 . . . . . 6 (𝐶𝐷𝐷 ∈ V)
4746ad3antlr 482 . . . . 5 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → 𝐷 ∈ V)
48 djuex 6894 . . . . . 6 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
49 brdomg 6608 . . . . . 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 406 . . . 4 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → ((𝐴𝐶) ≼ (𝐵𝐷) ↔ ∃ :(𝐴𝐶)–1-1→(𝐵𝐷)))
5243, 51mpbird 166 . . 3 ((((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) ∧ 𝑔:𝐶1-1𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))
534, 52exlimddv 1852 . 2 (((𝐴𝐵𝐶𝐷) ∧ 𝑓:𝐴1-1𝐵) → (𝐴𝐶) ≼ (𝐵𝐷))
542, 53exlimddv 1852 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cin 3038  wss 3039  c0 3331   class class class wbr 3897  ran crn 4508  cres 4509  ccom 4511  wf 5087  1-1wf1 5088  cdom 6599  cdju 6888  inlcinl 6896  inrcinr 6897  casecdjucase 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dom 6602  df-dju 6889  df-inl 6898  df-inr 6899  df-case 6935
This theorem is referenced by:  exmidfodomrlemr  7022  exmidfodomrlemrALT  7023  sbthom  13055
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