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Theorem updjudhcoinlf 9337
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 9336 . . . 4 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
54ffnd 6488 . . 3 (𝜑𝐻 Fn (𝐴𝐵))
6 inlresf 9319 . . . 4 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
7 ffn 6487 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → (inl ↾ 𝐴) Fn 𝐴)
86, 7mp1i 13 . . 3 (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
9 frn 6493 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
106, 9mp1i 13 . . 3 (𝜑 → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
11 fnco 6438 . . 3 ((𝐻 Fn (𝐴𝐵) ∧ (inl ↾ 𝐴) Fn 𝐴 ∧ ran (inl ↾ 𝐴) ⊆ (𝐴𝐵)) → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
125, 8, 10, 11syl3anc 1368 . 2 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
131ffnd 6488 . 2 (𝜑𝐹 Fn 𝐴)
14 fvco2 6731 . . . 4 (((inl ↾ 𝐴) Fn 𝐴𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
158, 14sylan 583 . . 3 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
16 fvres 6662 . . . . . 6 (𝑎𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
1716adantl 485 . . . . 5 ((𝜑𝑎𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
1817fveq2d 6647 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
19 fveqeq2 6652 . . . . . . . 8 (𝑥 = (inl‘𝑎) → ((1st𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
20 2fveq3 6648 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐹‘(2nd𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
21 2fveq3 6648 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐺‘(2nd𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
2219, 20, 21ifbieq12d 4467 . . . . . . 7 (𝑥 = (inl‘𝑎) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
2322adantl 485 . . . . . 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 9332 . . . . . . . . 9 (𝑎𝐴 → (1st ‘(inl‘𝑎)) = ∅)
2524adantl 485 . . . . . . . 8 ((𝜑𝑎𝐴) → (1st ‘(inl‘𝑎)) = ∅)
2625adantr 484 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
2726iftrued 4448 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))) = (𝐹‘(2nd ‘(inl‘𝑎))))
2823, 27eqtrd 2856 . . . . 5 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
29 djulcl 9315 . . . . . 6 (𝑎𝐴 → (inl‘𝑎) ∈ (𝐴𝐵))
3029adantl 485 . . . . 5 ((𝜑𝑎𝐴) → (inl‘𝑎) ∈ (𝐴𝐵))
311adantr 484 . . . . . 6 ((𝜑𝑎𝐴) → 𝐹:𝐴𝐶)
32 2ndinl 9333 . . . . . . . 8 (𝑎𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
3332adantl 485 . . . . . . 7 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
34 simpr 488 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐴)
3533, 34eqeltrd 2912 . . . . . 6 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
3631, 35ffvelrnd 6825 . . . . 5 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
373, 28, 30, 36fvmptd2 6749 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
3818, 37eqtrd 2856 . . 3 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
3933fveq2d 6647 . . 3 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹𝑎))
4015, 38, 393eqtrd 2860 . 2 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹𝑎))
4112, 13, 40eqfnfvd 6778 1 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wss 3910  c0 4266  ifcif 4440  cmpt 5119  ran crn 5529  cres 5530  ccom 5532   Fn wfn 6323  wf 6324  cfv 6328  1st c1st 7662  2nd c2nd 7663  cdju 9303  inlcinl 9304
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-1st 7664  df-2nd 7665  df-1o 8077  df-dju 9306  df-inl 9307
This theorem is referenced by:  updjud  9339
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