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Theorem updjudhcoinlf 9970
Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first 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
updjudhcoinlf (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝐹
Allowed substitution hints:   𝐺(𝑥)   𝐻(𝑥)

Proof of Theorem updjudhcoinlf
Dummy variable 𝑎 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 9969 . . . 4 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
54ffnd 6738 . . 3 (𝜑𝐻 Fn (𝐴𝐵))
6 inlresf 9952 . . . 4 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
7 ffn 6737 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → (inl ↾ 𝐴) Fn 𝐴)
86, 7mp1i 13 . . 3 (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
9 frn 6744 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
106, 9mp1i 13 . . 3 (𝜑 → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
11 fnco 6687 . . 3 ((𝐻 Fn (𝐴𝐵) ∧ (inl ↾ 𝐴) Fn 𝐴 ∧ ran (inl ↾ 𝐴) ⊆ (𝐴𝐵)) → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
125, 8, 10, 11syl3anc 1370 . 2 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
131ffnd 6738 . 2 (𝜑𝐹 Fn 𝐴)
14 fvco2 7006 . . . 4 (((inl ↾ 𝐴) Fn 𝐴𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
158, 14sylan 580 . . 3 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
16 fvres 6926 . . . . . 6 (𝑎𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
1716adantl 481 . . . . 5 ((𝜑𝑎𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
1817fveq2d 6911 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
19 fveqeq2 6916 . . . . . . . 8 (𝑥 = (inl‘𝑎) → ((1st𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
20 2fveq3 6912 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐹‘(2nd𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
21 2fveq3 6912 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐺‘(2nd𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
2219, 20, 21ifbieq12d 4559 . . . . . . 7 (𝑥 = (inl‘𝑎) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
2322adantl 481 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
24 1stinl 9965 . . . . . . . . 9 (𝑎𝐴 → (1st ‘(inl‘𝑎)) = ∅)
2524adantl 481 . . . . . . . 8 ((𝜑𝑎𝐴) → (1st ‘(inl‘𝑎)) = ∅)
2625adantr 480 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
2726iftrued 4539 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))) = (𝐹‘(2nd ‘(inl‘𝑎))))
2823, 27eqtrd 2775 . . . . 5 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
29 djulcl 9948 . . . . . 6 (𝑎𝐴 → (inl‘𝑎) ∈ (𝐴𝐵))
3029adantl 481 . . . . 5 ((𝜑𝑎𝐴) → (inl‘𝑎) ∈ (𝐴𝐵))
311adantr 480 . . . . . 6 ((𝜑𝑎𝐴) → 𝐹:𝐴𝐶)
32 2ndinl 9966 . . . . . . . 8 (𝑎𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
3332adantl 481 . . . . . . 7 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
34 simpr 484 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐴)
3533, 34eqeltrd 2839 . . . . . 6 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
3631, 35ffvelcdmd 7105 . . . . 5 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
373, 28, 30, 36fvmptd2 7024 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
3818, 37eqtrd 2775 . . 3 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
3933fveq2d 6911 . . 3 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹𝑎))
4015, 38, 393eqtrd 2779 . 2 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹𝑎))
4112, 13, 40eqfnfvd 7054 1 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963  c0 4339  ifcif 4531  cmpt 5231  ran crn 5690  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  1st c1st 8011  2nd c2nd 8012  cdju 9936  inlcinl 9937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014  df-1o 8505  df-dju 9939  df-inl 9940
This theorem is referenced by:  updjud  9972
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