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Theorem updjudhcoinrg 9874
Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
Hypotheses
Ref Expression
updjud.f (πœ‘ β†’ 𝐹:𝐴⟢𝐢)
updjud.g (πœ‘ β†’ 𝐺:𝐡⟢𝐢)
updjudhf.h 𝐻 = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))
Assertion
Ref Expression
updjudhcoinrg (πœ‘ β†’ (𝐻 ∘ (inr β†Ύ 𝐡)) = 𝐺)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢   πœ‘,π‘₯   π‘₯,𝐹   π‘₯,𝐺
Allowed substitution hint:   𝐻(π‘₯)

Proof of Theorem updjudhcoinrg
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 updjud.f . . . . 5 (πœ‘ β†’ 𝐹:𝐴⟢𝐢)
2 updjud.g . . . . 5 (πœ‘ β†’ 𝐺:𝐡⟢𝐢)
3 updjudhf.h . . . . 5 𝐻 = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))
41, 2, 3updjudhf 9872 . . . 4 (πœ‘ β†’ 𝐻:(𝐴 βŠ” 𝐡)⟢𝐢)
54ffnd 6670 . . 3 (πœ‘ β†’ 𝐻 Fn (𝐴 βŠ” 𝐡))
6 inrresf 9857 . . . 4 (inr β†Ύ 𝐡):𝐡⟢(𝐴 βŠ” 𝐡)
7 ffn 6669 . . . 4 ((inr β†Ύ 𝐡):𝐡⟢(𝐴 βŠ” 𝐡) β†’ (inr β†Ύ 𝐡) Fn 𝐡)
86, 7mp1i 13 . . 3 (πœ‘ β†’ (inr β†Ύ 𝐡) Fn 𝐡)
9 frn 6676 . . . 4 ((inr β†Ύ 𝐡):𝐡⟢(𝐴 βŠ” 𝐡) β†’ ran (inr β†Ύ 𝐡) βŠ† (𝐴 βŠ” 𝐡))
106, 9mp1i 13 . . 3 (πœ‘ β†’ ran (inr β†Ύ 𝐡) βŠ† (𝐴 βŠ” 𝐡))
11 fnco 6619 . . 3 ((𝐻 Fn (𝐴 βŠ” 𝐡) ∧ (inr β†Ύ 𝐡) Fn 𝐡 ∧ ran (inr β†Ύ 𝐡) βŠ† (𝐴 βŠ” 𝐡)) β†’ (𝐻 ∘ (inr β†Ύ 𝐡)) Fn 𝐡)
125, 8, 10, 11syl3anc 1372 . 2 (πœ‘ β†’ (𝐻 ∘ (inr β†Ύ 𝐡)) Fn 𝐡)
132ffnd 6670 . 2 (πœ‘ β†’ 𝐺 Fn 𝐡)
14 fvco2 6939 . . . 4 (((inr β†Ύ 𝐡) Fn 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ ((𝐻 ∘ (inr β†Ύ 𝐡))β€˜π‘) = (π»β€˜((inr β†Ύ 𝐡)β€˜π‘)))
158, 14sylan 581 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ((𝐻 ∘ (inr β†Ύ 𝐡))β€˜π‘) = (π»β€˜((inr β†Ύ 𝐡)β€˜π‘)))
16 fvres 6862 . . . . . 6 (𝑏 ∈ 𝐡 β†’ ((inr β†Ύ 𝐡)β€˜π‘) = (inrβ€˜π‘))
1716adantl 483 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ((inr β†Ύ 𝐡)β€˜π‘) = (inrβ€˜π‘))
1817fveq2d 6847 . . . 4 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (π»β€˜((inr β†Ύ 𝐡)β€˜π‘)) = (π»β€˜(inrβ€˜π‘)))
19 fveqeq2 6852 . . . . . . . 8 (π‘₯ = (inrβ€˜π‘) β†’ ((1st β€˜π‘₯) = βˆ… ↔ (1st β€˜(inrβ€˜π‘)) = βˆ…))
20 2fveq3 6848 . . . . . . . 8 (π‘₯ = (inrβ€˜π‘) β†’ (πΉβ€˜(2nd β€˜π‘₯)) = (πΉβ€˜(2nd β€˜(inrβ€˜π‘))))
21 2fveq3 6848 . . . . . . . 8 (π‘₯ = (inrβ€˜π‘) β†’ (πΊβ€˜(2nd β€˜π‘₯)) = (πΊβ€˜(2nd β€˜(inrβ€˜π‘))))
2219, 20, 21ifbieq12d 4515 . . . . . . 7 (π‘₯ = (inrβ€˜π‘) β†’ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))) = if((1st β€˜(inrβ€˜π‘)) = βˆ…, (πΉβ€˜(2nd β€˜(inrβ€˜π‘))), (πΊβ€˜(2nd β€˜(inrβ€˜π‘)))))
2322adantl 483 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ π‘₯ = (inrβ€˜π‘)) β†’ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))) = if((1st β€˜(inrβ€˜π‘)) = βˆ…, (πΉβ€˜(2nd β€˜(inrβ€˜π‘))), (πΊβ€˜(2nd β€˜(inrβ€˜π‘)))))
24 1stinr 9870 . . . . . . . . . 10 (𝑏 ∈ 𝐡 β†’ (1st β€˜(inrβ€˜π‘)) = 1o)
25 1n0 8435 . . . . . . . . . . . 12 1o β‰  βˆ…
2625neii 2942 . . . . . . . . . . 11 Β¬ 1o = βˆ…
27 eqeq1 2737 . . . . . . . . . . 11 ((1st β€˜(inrβ€˜π‘)) = 1o β†’ ((1st β€˜(inrβ€˜π‘)) = βˆ… ↔ 1o = βˆ…))
2826, 27mtbiri 327 . . . . . . . . . 10 ((1st β€˜(inrβ€˜π‘)) = 1o β†’ Β¬ (1st β€˜(inrβ€˜π‘)) = βˆ…)
2924, 28syl 17 . . . . . . . . 9 (𝑏 ∈ 𝐡 β†’ Β¬ (1st β€˜(inrβ€˜π‘)) = βˆ…)
3029adantl 483 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ Β¬ (1st β€˜(inrβ€˜π‘)) = βˆ…)
3130adantr 482 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ π‘₯ = (inrβ€˜π‘)) β†’ Β¬ (1st β€˜(inrβ€˜π‘)) = βˆ…)
3231iffalsed 4498 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ π‘₯ = (inrβ€˜π‘)) β†’ if((1st β€˜(inrβ€˜π‘)) = βˆ…, (πΉβ€˜(2nd β€˜(inrβ€˜π‘))), (πΊβ€˜(2nd β€˜(inrβ€˜π‘)))) = (πΊβ€˜(2nd β€˜(inrβ€˜π‘))))
3323, 32eqtrd 2773 . . . . 5 (((πœ‘ ∧ 𝑏 ∈ 𝐡) ∧ π‘₯ = (inrβ€˜π‘)) β†’ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))) = (πΊβ€˜(2nd β€˜(inrβ€˜π‘))))
34 djurcl 9852 . . . . . 6 (𝑏 ∈ 𝐡 β†’ (inrβ€˜π‘) ∈ (𝐴 βŠ” 𝐡))
3534adantl 483 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (inrβ€˜π‘) ∈ (𝐴 βŠ” 𝐡))
362adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝐺:𝐡⟢𝐢)
37 2ndinr 9871 . . . . . . . 8 (𝑏 ∈ 𝐡 β†’ (2nd β€˜(inrβ€˜π‘)) = 𝑏)
3837adantl 483 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (2nd β€˜(inrβ€˜π‘)) = 𝑏)
39 simpr 486 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ 𝐡)
4038, 39eqeltrd 2834 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (2nd β€˜(inrβ€˜π‘)) ∈ 𝐡)
4136, 40ffvelcdmd 7037 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (πΊβ€˜(2nd β€˜(inrβ€˜π‘))) ∈ 𝐢)
423, 33, 35, 41fvmptd2 6957 . . . 4 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (π»β€˜(inrβ€˜π‘)) = (πΊβ€˜(2nd β€˜(inrβ€˜π‘))))
4318, 42eqtrd 2773 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (π»β€˜((inr β†Ύ 𝐡)β€˜π‘)) = (πΊβ€˜(2nd β€˜(inrβ€˜π‘))))
4438fveq2d 6847 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (πΊβ€˜(2nd β€˜(inrβ€˜π‘))) = (πΊβ€˜π‘))
4515, 43, 443eqtrd 2777 . 2 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ((𝐻 ∘ (inr β†Ύ 𝐡))β€˜π‘) = (πΊβ€˜π‘))
4612, 13, 45eqfnfvd 6986 1 (πœ‘ β†’ (𝐻 ∘ (inr β†Ύ 𝐡)) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487   ↦ cmpt 5189  ran crn 5635   β†Ύ cres 5636   ∘ ccom 5638   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  1st c1st 7920  2nd c2nd 7921  1oc1o 8406   βŠ” cdju 9839  inrcinr 9841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1st 7922  df-2nd 7923  df-1o 8413  df-dju 9842  df-inr 9844
This theorem is referenced by:  updjud  9875
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