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Theorem updjudhcoinrg 9923
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 (𝜑𝐹:𝐴𝐶)
updjud.g (𝜑𝐺:𝐵𝐶)
updjudhf.h 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
Assertion
Ref Expression
updjudhcoinrg (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem updjudhcoinrg
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 updjud.f . . . . 5 (𝜑𝐹:𝐴𝐶)
2 updjud.g . . . . 5 (𝜑𝐺:𝐵𝐶)
3 updjudhf.h . . . . 5 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
41, 2, 3updjudhf 9921 . . . 4 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
54ffnd 6714 . . 3 (𝜑𝐻 Fn (𝐴𝐵))
6 inrresf 9906 . . . 4 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
7 ffn 6713 . . . 4 ((inr ↾ 𝐵):𝐵⟶(𝐴𝐵) → (inr ↾ 𝐵) Fn 𝐵)
86, 7mp1i 13 . . 3 (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
9 frn 6720 . . . 4 ((inr ↾ 𝐵):𝐵⟶(𝐴𝐵) → ran (inr ↾ 𝐵) ⊆ (𝐴𝐵))
106, 9mp1i 13 . . 3 (𝜑 → ran (inr ↾ 𝐵) ⊆ (𝐴𝐵))
11 fnco 6663 . . 3 ((𝐻 Fn (𝐴𝐵) ∧ (inr ↾ 𝐵) Fn 𝐵 ∧ ran (inr ↾ 𝐵) ⊆ (𝐴𝐵)) → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
125, 8, 10, 11syl3anc 1372 . 2 (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
132ffnd 6714 . 2 (𝜑𝐺 Fn 𝐵)
14 fvco2 6983 . . . 4 (((inr ↾ 𝐵) Fn 𝐵𝑏𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
158, 14sylan 581 . . 3 ((𝜑𝑏𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
16 fvres 6906 . . . . . 6 (𝑏𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
1716adantl 483 . . . . 5 ((𝜑𝑏𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
1817fveq2d 6891 . . . 4 ((𝜑𝑏𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
19 fveqeq2 6896 . . . . . . . 8 (𝑥 = (inr‘𝑏) → ((1st𝑥) = ∅ ↔ (1st ‘(inr‘𝑏)) = ∅))
20 2fveq3 6892 . . . . . . . 8 (𝑥 = (inr‘𝑏) → (𝐹‘(2nd𝑥)) = (𝐹‘(2nd ‘(inr‘𝑏))))
21 2fveq3 6892 . . . . . . . 8 (𝑥 = (inr‘𝑏) → (𝐺‘(2nd𝑥)) = (𝐺‘(2nd ‘(inr‘𝑏))))
2219, 20, 21ifbieq12d 4554 . . . . . . 7 (𝑥 = (inr‘𝑏) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))))
2322adantl 483 . . . . . 6 (((𝜑𝑏𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))))
24 1stinr 9919 . . . . . . . . . 10 (𝑏𝐵 → (1st ‘(inr‘𝑏)) = 1o)
25 1n0 8482 . . . . . . . . . . . 12 1o ≠ ∅
2625neii 2943 . . . . . . . . . . 11 ¬ 1o = ∅
27 eqeq1 2737 . . . . . . . . . . 11 ((1st ‘(inr‘𝑏)) = 1o → ((1st ‘(inr‘𝑏)) = ∅ ↔ 1o = ∅))
2826, 27mtbiri 327 . . . . . . . . . 10 ((1st ‘(inr‘𝑏)) = 1o → ¬ (1st ‘(inr‘𝑏)) = ∅)
2924, 28syl 17 . . . . . . . . 9 (𝑏𝐵 → ¬ (1st ‘(inr‘𝑏)) = ∅)
3029adantl 483 . . . . . . . 8 ((𝜑𝑏𝐵) → ¬ (1st ‘(inr‘𝑏)) = ∅)
3130adantr 482 . . . . . . 7 (((𝜑𝑏𝐵) ∧ 𝑥 = (inr‘𝑏)) → ¬ (1st ‘(inr‘𝑏)) = ∅)
3231iffalsed 4537 . . . . . 6 (((𝜑𝑏𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))) = (𝐺‘(2nd ‘(inr‘𝑏))))
3323, 32eqtrd 2773 . . . . 5 (((𝜑𝑏𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = (𝐺‘(2nd ‘(inr‘𝑏))))
34 djurcl 9901 . . . . . 6 (𝑏𝐵 → (inr‘𝑏) ∈ (𝐴𝐵))
3534adantl 483 . . . . 5 ((𝜑𝑏𝐵) → (inr‘𝑏) ∈ (𝐴𝐵))
362adantr 482 . . . . . 6 ((𝜑𝑏𝐵) → 𝐺:𝐵𝐶)
37 2ndinr 9920 . . . . . . . 8 (𝑏𝐵 → (2nd ‘(inr‘𝑏)) = 𝑏)
3837adantl 483 . . . . . . 7 ((𝜑𝑏𝐵) → (2nd ‘(inr‘𝑏)) = 𝑏)
39 simpr 486 . . . . . . 7 ((𝜑𝑏𝐵) → 𝑏𝐵)
4038, 39eqeltrd 2834 . . . . . 6 ((𝜑𝑏𝐵) → (2nd ‘(inr‘𝑏)) ∈ 𝐵)
4136, 40ffvelcdmd 7082 . . . . 5 ((𝜑𝑏𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) ∈ 𝐶)
423, 33, 35, 41fvmptd2 7001 . . . 4 ((𝜑𝑏𝐵) → (𝐻‘(inr‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
4318, 42eqtrd 2773 . . 3 ((𝜑𝑏𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
4438fveq2d 6891 . . 3 ((𝜑𝑏𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) = (𝐺𝑏))
4515, 43, 443eqtrd 2777 . 2 ((𝜑𝑏𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐺𝑏))
4612, 13, 45eqfnfvd 7030 1 (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3946  c0 4320  ifcif 4526  cmpt 5229  ran crn 5675  cres 5676  ccom 5678   Fn wfn 6534  wf 6535  cfv 6539  1st c1st 7967  2nd c2nd 7968  1oc1o 8453  cdju 9888  inrcinr 9890
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 5297  ax-nul 5304  ax-pr 5425  ax-un 7719
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-om 7850  df-1st 7969  df-2nd 7970  df-1o 8460  df-dju 9891  df-inr 9893
This theorem is referenced by:  updjud  9924
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