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Theorem updjudhcoinlf 9873
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 (πœ‘ β†’ 𝐹:𝐴⟢𝐢)
updjud.g (πœ‘ β†’ 𝐺:𝐡⟢𝐢)
updjudhf.h 𝐻 = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))
Assertion
Ref Expression
updjudhcoinlf (πœ‘ β†’ (𝐻 ∘ (inl β†Ύ 𝐴)) = 𝐹)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢   πœ‘,π‘₯   π‘₯,𝐹
Allowed substitution hints:   𝐺(π‘₯)   𝐻(π‘₯)

Proof of Theorem updjudhcoinlf
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 updjud.f . . . . 5 (πœ‘ β†’ 𝐹:𝐴⟢𝐢)
2 updjud.g . . . . 5 (πœ‘ β†’ 𝐺:𝐡⟢𝐢)
3 updjudhf.h . . . . 5 𝐻 = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))
41, 2, 3updjudhf 9872 . . . 4 (πœ‘ β†’ 𝐻:(𝐴 βŠ” 𝐡)⟢𝐢)
54ffnd 6670 . . 3 (πœ‘ β†’ 𝐻 Fn (𝐴 βŠ” 𝐡))
6 inlresf 9855 . . . 4 (inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡)
7 ffn 6669 . . . 4 ((inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡) β†’ (inl β†Ύ 𝐴) Fn 𝐴)
86, 7mp1i 13 . . 3 (πœ‘ β†’ (inl β†Ύ 𝐴) Fn 𝐴)
9 frn 6676 . . . 4 ((inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡) β†’ ran (inl β†Ύ 𝐴) βŠ† (𝐴 βŠ” 𝐡))
106, 9mp1i 13 . . 3 (πœ‘ β†’ ran (inl β†Ύ 𝐴) βŠ† (𝐴 βŠ” 𝐡))
11 fnco 6619 . . 3 ((𝐻 Fn (𝐴 βŠ” 𝐡) ∧ (inl β†Ύ 𝐴) Fn 𝐴 ∧ ran (inl β†Ύ 𝐴) βŠ† (𝐴 βŠ” 𝐡)) β†’ (𝐻 ∘ (inl β†Ύ 𝐴)) Fn 𝐴)
125, 8, 10, 11syl3anc 1372 . 2 (πœ‘ β†’ (𝐻 ∘ (inl β†Ύ 𝐴)) Fn 𝐴)
131ffnd 6670 . 2 (πœ‘ β†’ 𝐹 Fn 𝐴)
14 fvco2 6939 . . . 4 (((inl β†Ύ 𝐴) Fn 𝐴 ∧ π‘Ž ∈ 𝐴) β†’ ((𝐻 ∘ (inl β†Ύ 𝐴))β€˜π‘Ž) = (π»β€˜((inl β†Ύ 𝐴)β€˜π‘Ž)))
158, 14sylan 581 . . 3 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ ((𝐻 ∘ (inl β†Ύ 𝐴))β€˜π‘Ž) = (π»β€˜((inl β†Ύ 𝐴)β€˜π‘Ž)))
16 fvres 6862 . . . . . 6 (π‘Ž ∈ 𝐴 β†’ ((inl β†Ύ 𝐴)β€˜π‘Ž) = (inlβ€˜π‘Ž))
1716adantl 483 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ ((inl β†Ύ 𝐴)β€˜π‘Ž) = (inlβ€˜π‘Ž))
1817fveq2d 6847 . . . 4 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π»β€˜((inl β†Ύ 𝐴)β€˜π‘Ž)) = (π»β€˜(inlβ€˜π‘Ž)))
19 fveqeq2 6852 . . . . . . . 8 (π‘₯ = (inlβ€˜π‘Ž) β†’ ((1st β€˜π‘₯) = βˆ… ↔ (1st β€˜(inlβ€˜π‘Ž)) = βˆ…))
20 2fveq3 6848 . . . . . . . 8 (π‘₯ = (inlβ€˜π‘Ž) β†’ (πΉβ€˜(2nd β€˜π‘₯)) = (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))))
21 2fveq3 6848 . . . . . . . 8 (π‘₯ = (inlβ€˜π‘Ž) β†’ (πΊβ€˜(2nd β€˜π‘₯)) = (πΊβ€˜(2nd β€˜(inlβ€˜π‘Ž))))
2219, 20, 21ifbieq12d 4515 . . . . . . 7 (π‘₯ = (inlβ€˜π‘Ž) β†’ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))) = if((1st β€˜(inlβ€˜π‘Ž)) = βˆ…, (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))), (πΊβ€˜(2nd β€˜(inlβ€˜π‘Ž)))))
2322adantl 483 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘Ž)) β†’ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))) = if((1st β€˜(inlβ€˜π‘Ž)) = βˆ…, (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))), (πΊβ€˜(2nd β€˜(inlβ€˜π‘Ž)))))
24 1stinl 9868 . . . . . . . . 9 (π‘Ž ∈ 𝐴 β†’ (1st β€˜(inlβ€˜π‘Ž)) = βˆ…)
2524adantl 483 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (1st β€˜(inlβ€˜π‘Ž)) = βˆ…)
2625adantr 482 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘Ž)) β†’ (1st β€˜(inlβ€˜π‘Ž)) = βˆ…)
2726iftrued 4495 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘Ž)) β†’ if((1st β€˜(inlβ€˜π‘Ž)) = βˆ…, (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))), (πΊβ€˜(2nd β€˜(inlβ€˜π‘Ž)))) = (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))))
2823, 27eqtrd 2773 . . . . 5 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘Ž)) β†’ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))) = (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))))
29 djulcl 9851 . . . . . 6 (π‘Ž ∈ 𝐴 β†’ (inlβ€˜π‘Ž) ∈ (𝐴 βŠ” 𝐡))
3029adantl 483 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (inlβ€˜π‘Ž) ∈ (𝐴 βŠ” 𝐡))
311adantr 482 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ 𝐹:𝐴⟢𝐢)
32 2ndinl 9869 . . . . . . . 8 (π‘Ž ∈ 𝐴 β†’ (2nd β€˜(inlβ€˜π‘Ž)) = π‘Ž)
3332adantl 483 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (2nd β€˜(inlβ€˜π‘Ž)) = π‘Ž)
34 simpr 486 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž ∈ 𝐴)
3533, 34eqeltrd 2834 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (2nd β€˜(inlβ€˜π‘Ž)) ∈ 𝐴)
3631, 35ffvelcdmd 7037 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))) ∈ 𝐢)
373, 28, 30, 36fvmptd2 6957 . . . 4 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π»β€˜(inlβ€˜π‘Ž)) = (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))))
3818, 37eqtrd 2773 . . 3 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (π»β€˜((inl β†Ύ 𝐴)β€˜π‘Ž)) = (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))))
3933fveq2d 6847 . . 3 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜(2nd β€˜(inlβ€˜π‘Ž))) = (πΉβ€˜π‘Ž))
4015, 38, 393eqtrd 2777 . 2 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ ((𝐻 ∘ (inl β†Ύ 𝐴))β€˜π‘Ž) = (πΉβ€˜π‘Ž))
4112, 13, 40eqfnfvd 6986 1 (πœ‘ β†’ (𝐻 ∘ (inl β†Ύ 𝐴)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487   ↦ cmpt 5189  ran crn 5635   β†Ύ cres 5636   ∘ ccom 5638   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  1st c1st 7920  2nd c2nd 7921   βŠ” cdju 9839  inlcinl 9840
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 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-1st 7922  df-2nd 7923  df-1o 8413  df-dju 9842  df-inl 9843
This theorem is referenced by:  updjud  9875
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