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Theorem updjudhcoinlf 9972
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 9971 . . . 4 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
54ffnd 6737 . . 3 (𝜑𝐻 Fn (𝐴𝐵))
6 inlresf 9954 . . . 4 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
7 ffn 6736 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → (inl ↾ 𝐴) Fn 𝐴)
86, 7mp1i 13 . . 3 (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
9 frn 6743 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
106, 9mp1i 13 . . 3 (𝜑 → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
11 fnco 6686 . . 3 ((𝐻 Fn (𝐴𝐵) ∧ (inl ↾ 𝐴) Fn 𝐴 ∧ ran (inl ↾ 𝐴) ⊆ (𝐴𝐵)) → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
125, 8, 10, 11syl3anc 1373 . 2 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
131ffnd 6737 . 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 6925 . . . . . 6 (𝑎𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
1716adantl 481 . . . . 5 ((𝜑𝑎𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
1817fveq2d 6910 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
19 fveqeq2 6915 . . . . . . . 8 (𝑥 = (inl‘𝑎) → ((1st𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
20 2fveq3 6911 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐹‘(2nd𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
21 2fveq3 6911 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐺‘(2nd𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
2219, 20, 21ifbieq12d 4554 . . . . . . 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 9967 . . . . . . . . 9 (𝑎𝐴 → (1st ‘(inl‘𝑎)) = ∅)
2524adantl 481 . . . . . . . 8 ((𝜑𝑎𝐴) → (1st ‘(inl‘𝑎)) = ∅)
2625adantr 480 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
2726iftrued 4533 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))) = (𝐹‘(2nd ‘(inl‘𝑎))))
2823, 27eqtrd 2777 . . . . 5 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
29 djulcl 9950 . . . . . 6 (𝑎𝐴 → (inl‘𝑎) ∈ (𝐴𝐵))
3029adantl 481 . . . . 5 ((𝜑𝑎𝐴) → (inl‘𝑎) ∈ (𝐴𝐵))
311adantr 480 . . . . . 6 ((𝜑𝑎𝐴) → 𝐹:𝐴𝐶)
32 2ndinl 9968 . . . . . . . 8 (𝑎𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
3332adantl 481 . . . . . . 7 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
34 simpr 484 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐴)
3533, 34eqeltrd 2841 . . . . . 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 2777 . . 3 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
3933fveq2d 6910 . . 3 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹𝑎))
4015, 38, 393eqtrd 2781 . 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 1540  wcel 2108  wss 3951  c0 4333  ifcif 4525  cmpt 5225  ran crn 5686  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  1st c1st 8012  2nd c2nd 8013  cdju 9938  inlcinl 9939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-dju 9941  df-inl 9942
This theorem is referenced by:  updjud  9974
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