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Theorem updjudhcoinlf 6931
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 6930 . . . 4 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
5 ffn 5240 . . . 4 (𝐻:(𝐴𝐵)⟶𝐶𝐻 Fn (𝐴𝐵))
64, 5syl 14 . . 3 (𝜑𝐻 Fn (𝐴𝐵))
7 inlresf1 6912 . . . 4 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
8 f1fn 5298 . . . 4 ((inl ↾ 𝐴):𝐴1-1→(𝐴𝐵) → (inl ↾ 𝐴) Fn 𝐴)
97, 8mp1i 10 . . 3 (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
10 f1f 5296 . . . . 5 ((inl ↾ 𝐴):𝐴1-1→(𝐴𝐵) → (inl ↾ 𝐴):𝐴⟶(𝐴𝐵))
117, 10ax-mp 5 . . . 4 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
12 frn 5249 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
1311, 12mp1i 10 . . 3 (𝜑 → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
14 fnco 5199 . . 3 ((𝐻 Fn (𝐴𝐵) ∧ (inl ↾ 𝐴) Fn 𝐴 ∧ ran (inl ↾ 𝐴) ⊆ (𝐴𝐵)) → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
156, 9, 13, 14syl3anc 1199 . 2 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
16 ffn 5240 . . 3 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
171, 16syl 14 . 2 (𝜑𝐹 Fn 𝐴)
18 fvco2 5456 . . . 4 (((inl ↾ 𝐴) Fn 𝐴𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
199, 18sylan 279 . . 3 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
20 fvres 5411 . . . . . 6 (𝑎𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
2120adantl 273 . . . . 5 ((𝜑𝑎𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
2221fveq2d 5391 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
233a1i 9 . . . . 5 ((𝜑𝑎𝐴) → 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))))
24 fveq2 5387 . . . . . . . . 9 (𝑥 = (inl‘𝑎) → (1st𝑥) = (1st ‘(inl‘𝑎)))
2524eqeq1d 2124 . . . . . . . 8 (𝑥 = (inl‘𝑎) → ((1st𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
26 fveq2 5387 . . . . . . . . 9 (𝑥 = (inl‘𝑎) → (2nd𝑥) = (2nd ‘(inl‘𝑎)))
2726fveq2d 5391 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐹‘(2nd𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
2826fveq2d 5391 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐺‘(2nd𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
2925, 27, 28ifbieq12d 3466 . . . . . . 7 (𝑥 = (inl‘𝑎) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
3029adantl 273 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
31 1stinl 6925 . . . . . . . . 9 (𝑎𝐴 → (1st ‘(inl‘𝑎)) = ∅)
3231adantl 273 . . . . . . . 8 ((𝜑𝑎𝐴) → (1st ‘(inl‘𝑎)) = ∅)
3332adantr 272 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
3433iftrued 3449 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))) = (𝐹‘(2nd ‘(inl‘𝑎))))
3530, 34eqtrd 2148 . . . . 5 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
36 djulcl 6902 . . . . . 6 (𝑎𝐴 → (inl‘𝑎) ∈ (𝐴𝐵))
3736adantl 273 . . . . 5 ((𝜑𝑎𝐴) → (inl‘𝑎) ∈ (𝐴𝐵))
381adantr 272 . . . . . 6 ((𝜑𝑎𝐴) → 𝐹:𝐴𝐶)
39 2ndinl 6926 . . . . . . . 8 (𝑎𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
4039adantl 273 . . . . . . 7 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
41 simpr 109 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐴)
4240, 41eqeltrd 2192 . . . . . 6 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
4338, 42ffvelrnd 5522 . . . . 5 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
4423, 35, 37, 43fvmptd 5468 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
4522, 44eqtrd 2148 . . 3 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
4640fveq2d 5391 . . 3 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹𝑎))
4719, 45, 463eqtrd 2152 . 2 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹𝑎))
4815, 17, 47eqfnfvd 5487 1 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  wss 3039  c0 3331  ifcif 3442  cmpt 3957  ran crn 4508  cres 4509  ccom 4511   Fn wfn 5086  wf 5087  1-1wf1 5088  cfv 5091  1st c1st 6002  2nd c2nd 6003  cdju 6888  inlcinl 6896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by:  updjud  6933
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