Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  updjudhcoinrg Structured version   Visualization version   GIF version

Theorem updjudhcoinrg 9349
 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 9347 . . . 4 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
54ffnd 6489 . . 3 (𝜑𝐻 Fn (𝐴𝐵))
6 inrresf 9332 . . . 4 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
7 ffn 6488 . . . 4 ((inr ↾ 𝐵):𝐵⟶(𝐴𝐵) → (inr ↾ 𝐵) Fn 𝐵)
86, 7mp1i 13 . . 3 (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
9 frn 6494 . . . 4 ((inr ↾ 𝐵):𝐵⟶(𝐴𝐵) → ran (inr ↾ 𝐵) ⊆ (𝐴𝐵))
106, 9mp1i 13 . . 3 (𝜑 → ran (inr ↾ 𝐵) ⊆ (𝐴𝐵))
11 fnco 6438 . . 3 ((𝐻 Fn (𝐴𝐵) ∧ (inr ↾ 𝐵) Fn 𝐵 ∧ ran (inr ↾ 𝐵) ⊆ (𝐴𝐵)) → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
125, 8, 10, 11syl3anc 1368 . 2 (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
132ffnd 6489 . 2 (𝜑𝐺 Fn 𝐵)
14 fvco2 6736 . . . 4 (((inr ↾ 𝐵) Fn 𝐵𝑏𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
158, 14sylan 583 . . 3 ((𝜑𝑏𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
16 fvres 6665 . . . . . 6 (𝑏𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
1716adantl 485 . . . . 5 ((𝜑𝑏𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
1817fveq2d 6650 . . . 4 ((𝜑𝑏𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
19 fveqeq2 6655 . . . . . . . 8 (𝑥 = (inr‘𝑏) → ((1st𝑥) = ∅ ↔ (1st ‘(inr‘𝑏)) = ∅))
20 2fveq3 6651 . . . . . . . 8 (𝑥 = (inr‘𝑏) → (𝐹‘(2nd𝑥)) = (𝐹‘(2nd ‘(inr‘𝑏))))
21 2fveq3 6651 . . . . . . . 8 (𝑥 = (inr‘𝑏) → (𝐺‘(2nd𝑥)) = (𝐺‘(2nd ‘(inr‘𝑏))))
2219, 20, 21ifbieq12d 4452 . . . . . . 7 (𝑥 = (inr‘𝑏) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))))
2322adantl 485 . . . . . 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 9345 . . . . . . . . . 10 (𝑏𝐵 → (1st ‘(inr‘𝑏)) = 1o)
25 1n0 8105 . . . . . . . . . . . 12 1o ≠ ∅
2625neii 2989 . . . . . . . . . . 11 ¬ 1o = ∅
27 eqeq1 2802 . . . . . . . . . . 11 ((1st ‘(inr‘𝑏)) = 1o → ((1st ‘(inr‘𝑏)) = ∅ ↔ 1o = ∅))
2826, 27mtbiri 330 . . . . . . . . . 10 ((1st ‘(inr‘𝑏)) = 1o → ¬ (1st ‘(inr‘𝑏)) = ∅)
2924, 28syl 17 . . . . . . . . 9 (𝑏𝐵 → ¬ (1st ‘(inr‘𝑏)) = ∅)
3029adantl 485 . . . . . . . 8 ((𝜑𝑏𝐵) → ¬ (1st ‘(inr‘𝑏)) = ∅)
3130adantr 484 . . . . . . 7 (((𝜑𝑏𝐵) ∧ 𝑥 = (inr‘𝑏)) → ¬ (1st ‘(inr‘𝑏)) = ∅)
3231iffalsed 4436 . . . . . 6 (((𝜑𝑏𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))) = (𝐺‘(2nd ‘(inr‘𝑏))))
3323, 32eqtrd 2833 . . . . 5 (((𝜑𝑏𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = (𝐺‘(2nd ‘(inr‘𝑏))))
34 djurcl 9327 . . . . . 6 (𝑏𝐵 → (inr‘𝑏) ∈ (𝐴𝐵))
3534adantl 485 . . . . 5 ((𝜑𝑏𝐵) → (inr‘𝑏) ∈ (𝐴𝐵))
362adantr 484 . . . . . 6 ((𝜑𝑏𝐵) → 𝐺:𝐵𝐶)
37 2ndinr 9346 . . . . . . . 8 (𝑏𝐵 → (2nd ‘(inr‘𝑏)) = 𝑏)
3837adantl 485 . . . . . . 7 ((𝜑𝑏𝐵) → (2nd ‘(inr‘𝑏)) = 𝑏)
39 simpr 488 . . . . . . 7 ((𝜑𝑏𝐵) → 𝑏𝐵)
4038, 39eqeltrd 2890 . . . . . 6 ((𝜑𝑏𝐵) → (2nd ‘(inr‘𝑏)) ∈ 𝐵)
4136, 40ffvelrnd 6830 . . . . 5 ((𝜑𝑏𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) ∈ 𝐶)
423, 33, 35, 41fvmptd2 6754 . . . 4 ((𝜑𝑏𝐵) → (𝐻‘(inr‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
4318, 42eqtrd 2833 . . 3 ((𝜑𝑏𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
4438fveq2d 6650 . . 3 ((𝜑𝑏𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) = (𝐺𝑏))
4515, 43, 443eqtrd 2837 . 2 ((𝜑𝑏𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐺𝑏))
4612, 13, 45eqfnfvd 6783 1 (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∅c0 4243  ifcif 4425   ↦ cmpt 5111  ran crn 5521   ↾ cres 5522   ∘ ccom 5524   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  1st c1st 7672  2nd c2nd 7673  1oc1o 8081   ⊔ cdju 9314  inrcinr 9316 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-om 7564  df-1st 7674  df-2nd 7675  df-1o 8088  df-dju 9317  df-inr 9319 This theorem is referenced by:  updjud  9350
 Copyright terms: Public domain W3C validator